Lecture Notes 2 Random Variables. Discrete Random Variables: Probability mass function (pmf)
|
|
- Harriet Sutton
- 5 years ago
- Views:
Transcription
1 Lecture Notes 2 Random Variables Definition Discrete Random Variables: Probability mass function (pmf) Continuous Random Variables: Probability density function (pdf) Mean and Variance Cumulative Distribution Function (cdf) Functions of Random Variables Corresponding pages from B&T tetbook: 72 83, 86, 88, 90, 40 44, 46 50, 52 57, EE 78/278A: Random Variables Page 2 Random Variable A random variable is a real-valued variable that takes on values randomly Sounds nice, but not terribly precise or useful Mathematically, a random variable (r.v.) X is a real-valued function X(ω) over the sample space Ω of a random eperiment, i.e., X : Ω R Ω ω X(ω) Randomness comes from the fact that outcomes are random (X(ω) is a deterministic function) Notations: Always use upper case letters for random variables (X, Y,...) Always use lower case letters for values of random variables: X means that the random variable X takes on the value EE 78/278A: Random Variables Page 2 2
2 Eamples:. Flip a coin n times. Here Ω {H,T} n. Define the random variable X {0,,2,...,n} to be the number of heads 2. Roll a 4-sided die twice. (a) Define the random variable X as the maimum of the two rolls 4 3 2nd roll st roll Real Line EE 78/278A: Random Variables Page 2 3 (b) Define the random variable Y to be the sum of the outcomes of the two rolls (c) Define the random variable Z to be 0 if the sum of the two rolls is odd and if it is even 3. Flip coin until first heads shows up. Define the random variable X {,2...} to be the number of flips until the first heads 4. Let Ω R. Define the two random variables (a) X { ω + for ω 0 (b) Y otherwise 5. n packets arrive at a node in a communication network. Here Ω is the set of arrival time sequences (t,t 2,...,t n ) (0, ) n (a) Define the random variable N to be the number of packets arriving in the interval (0, ] (b) Define the random variable T to be the first interarrival time EE 78/278A: Random Variables Page 2 4
3 Why do we need random variables? Random variable can represent the gain or loss in a random eperiment, e.g., stock market Random variable can represent a measurement over a random eperiment, e.g., noise voltage on a resistor In most applications we care more about these costs/measurements than the underlying probability space Very often we work directly with random variables without knowing (or caring to know) the underlying probability space EE 78/278A: Random Variables Page 2 5 Specifying a Random Variable Specifying a random variable means being able to determine the probability that X A for any event A R, e.g., any interval To do so, we consider the inverse image of the set A under X(ω), {w : X(ω) A} R set A inverse image of A under X(ω), i.e., {ω : X(ω) A} So, X A iff ω {w : X(ω) A}, thus P({X A}) P({w : X(ω) A}), or in short P{X A} P{w : X(ω) A} EE 78/278A: Random Variables Page 2 6
4 Eample: Roll fair 4-sided die twice independently: Define the r.v. X to be the maimum of the two rolls. What is the P{0.5 < X 2}? 4 3 2nd roll st roll EE 78/278A: Random Variables Page 2 7 We classify r.v.s as: Discrete: X can assume only one of a countable number of values. Such r.v. can be specified by a Probability Mass Function (pmf). Eamples, 2, 3, 4(b), and 5(a) are of discrete r.v.s Continuous: X can assume one of a continuum of values and the probability of each value is 0. Such r.v. can be specified by a Probability Density Function (pdf). Eamples 4(a) and 5(b) are of continuous r.v.s Mied: X is neither discrete nor continuous. Such r.v. (as well as discrete and continuous r.v.s) can be specified by a Cumulative Distribution Function (cdf) EE 78/278A: Random Variables Page 2 8
5 Discrete Random Variables A random variable is said to be discrete if for some countable set X R, i.e., X {, 2,...}, P{X X} Eamples, 2, 3, 4(b), and 5(a) are discrete random variables Here X(ω) partitions Ω into the sets {ω : X(ω) i } for i,2,... Therefore, to specify X, it suffices to know P{X i } for all i R n... Ω A discrete random variable is thus completely specified by its probability mass function (pmf) p X () P{X } for all X EE 78/278A: Random Variables Page 2 9 Clearly p X () 0 and X p X() Eample: Roll a fair 4-sided die twice independently: Define the r.v. X to be the maimum of the two rolls 4 3 2nd roll st roll p X (): EE 78/278A: Random Variables Page 2 0
6 Note that p X () can be viewed as a probability measure over a discrete sample space (even though the original sample space may be continuous as in eamples 4(b) and 5(a)) The probability of any event A R is given by P{X A} A X p X () For the previous eample P{ < X 2.5 or X 3.5} Notation: We use X p X () or simply X p() to mean that the discrete random variable X has pmf p X () or p() EE 78/278A: Random Variables Page 2 Famous Probability Mass Functions Bernoulli: X Bern(p) for 0 p has pmf p X () p, and p X (0) p Geometric: X Geom(p) for 0 p has pmf p X (k) p( p) k for k,2,... p X (k) p k This r.v. represents, for eample, the number of coin flips until the first heads shows up (assuming independent coin flips) EE 78/278A: Random Variables Page 2 2
7 Binomial: X B(n,p) for integer n > 0 and 0 p has pmf ( ) n p X (k) p k ( p) (n k) for k 0,,2,...,n k p X (k) k k n The maimum of p X (k) is attained at { k (n+)p, (n+)p, if (n+)p is an integer [(n+)p], otherwise The binomial r.v. represents, for eample, the number of heads in n independent coin flips (see page 48 of Lecture Notes ) EE 78/278A: Random Variables Page 2 3 Poisson: X Poisson(λ) for λ > 0 (called the rate) has pmf p X (k) λk k! e λ for k 0,,2,... p X (k) The maimum of p X (k) attained at { k λ, λ, if λ is an integer [λ], otherwise k The Poisson r.v. often represents the number of random events, e.g., arrivals of packets, photons, customers, etc.in some time interval, e.g., (0, ] k EE 78/278A: Random Variables Page 2 4
8 Poisson is the limit of Binomial when p n, as n To show this let X n B(n, λ n ) for λ > 0. For any fied nonnegative integer k p Xn (k) ( n k )( λ n ) k ( λ ) (n k) n n(n )...(n k +) λ k k! n k n(n )...(n k +) λ k (n λ) k k! n(n )...(n k +) λ k (n λ)(n λ)...(n λ) k! λk k! e λ as n ( n λ n ( n λ ) n k ) n n ( λ n ) n EE 78/278A: Random Variables Page 2 5 Continuous Random Variables Suppose a r.v. X can take on a continuum of values each with probability 0 Eamples: Pick a number between 0 and Measure the voltage across a heated resistor Measure the phase of a random sinusoid... How do we describe probabilities of interesting events? Idea: For discrete r.v., we sum a pmf over points in a set to find its probability. For continuous r.v., integrate a probability density over a set to find its probability analogous to mass density in physics (integrate mass density to get the mass) EE 78/278A: Random Variables Page 2 6
9 Probability Density Function A continuous r.v. X can be specified by a probability density function f X () (pdf) such that, for any event A, P{X A} f X () d For eample, for A (a,b], the probability can be computed as Properties of f X ():. f X () 0 2. f X()d P{X (a,b]} A b a f X ()d Important note: f X () should not be interpreted as the probability that X, in fact it is not a probability measure, e.g., it can be > EE 78/278A: Random Variables Page 2 7 Can relate f X () to a probability using mean value theorem for integrals: Fi and some > 0. Then provided f X is continuous over (,+ ], P{X (,+ ]} Now, if is sufficently small, then + f X (α)dα f X (c) for some c + P{X (,+ ]} f X () Notation: X f X () means that X has pdf f X () EE 78/278A: Random Variables Page 2 8
10 Famous Probability Density Functions Uniform: X U[a,b] for b > a has the pdf f X () f() { b a for a b 0 otherwise b a a Uniform r.v. is commonly used to model quantization noise and finite precision computation error (roundoff error) b EE 78/278A: Random Variables Page 2 9 Eponential: X Ep(λ) for λ > 0 has the pdf f X () f() { λe λ for 0 0 otherwise λ Eponential r.v. is commonly used to model interarrival time in a queue, i.e., time between two consecutive packet or customer arrivals, service time in a queue, and lifetime of a particle, etc.. EE 78/278A: Random Variables Page 2 20
11 Eample: Let X Ep(0.) be the customer service time at a bank (in minutes). The person ahead of you has been served for 5 minutes. What is the probability that you will wait another 5 minutes or more before getting served? We want to find P{X > 0 X > 5} Solution: By definition of conditional probability P{X > 0 X > 5} P{X > 0, X > 5} P{X > 5} P{X > 0} P{X > 5} 0 0.e 0. d 5 0.e 0. d e e 0.5 e 0.5, but P{X > 5} e 0.5, i.e., the conditional probability of waiting more than 5 additional minutes given that you have already waited more than 5 minutes is the same as the unconditional probability of waiting more than 5 minutes! EE 78/278A: Random Variables Page 2 2 This is because the eponential r.v. is memoryless, which in general means that for any 0 <, To show this, consider P{X > X > } P{X > } P{X > X > } P{X >, X > } P{X > } P{X > } P{X > } λe λ d λe λ d e λ e λ e λ( ) P{X > } EE 78/278A: Random Variables Page 2 22
12 Gaussian: X N(µ,σ 2 ) has pdf f X () 2πσ 2 e ( µ)2 2σ 2 for < <, where µ is the mean and σ 2 is the variance N(µ,σ 2 ) µ Gaussian r.v.s are frequently encountered in nature, e.g., thermal and shot noise in electronic devices are Gaussian, and very frequently used in modelling various social, biological, and other phenomena EE 78/278A: Random Variables Page 2 23 Mean and Variance A discrete (continuous) r.v. is completely specified by its pmf (pdf) It is often desirable to summarize the r.v. or predict its outcome in terms of one or a few numbers. What do we epect the value of the r.v. to be? What range of values around the mean do we epect the r.v. to take? Such information can be provided by the mean and standard deviation of the r.v. First we consider discrete r.v.s Let X p X (). The epected value (or mean) of X is defined as E(X) X p X () Interpretations: If we view probabilities as relative frequencies, the mean would be the weighted sum of the relative frequencies. If we view probabilities as point masses, the mean would be the center of mass of the set of mass points EE 78/278A: Random Variables Page 2 24
13 Eample: If the weather is good, which happens with probability 0.6, Alice walks the 2 miles to class at a speed 5 miles/hr, otherwise she rides a bus at speed 30 miles/hr. What is the epected time to get to class? Solution: Define the discrete r.v. T to take the value (2/5)hr with probability 0.6 and (2/30)hr with probability 0.4. The epected value of T E(T) 2/ / /5 hr The second moment (or mean square or average power) of X is defined as E(X 2 ) X 2 p X () For the previous eample, the second moment is E(T 2 ) (2/5) (2/30) /225 hr 2 The variance of X is defined as Var(X) E [ (X E(X)) 2] X( E(X)) 2 p X () EE 78/278A: Random Variables Page 2 25 Note that the Var 0. The variance has the interpretation of the moment of inertia about the center of mass for a set of mass points The standard deviation of X is defined as σ X Var(X) Variance in terms of mean and second moment: Epanding the square and using the linearity of sum, we obtain Var(X) E [ (X E(X)) 2] ( E(X)) 2 p X () ( 2 2E(X)+[E(X)] 2 )p X () 2 p X () 2E(X) p X ()+[E(X)] 2 p X () E(X 2 ) 2[E(X)] 2 +[E(X)] 2 E(X 2 ) [E(X)] 2 Note that since for any r.v., Var(X) 0, E(X 2 ) (E(X)) 2 So, for our eample, Var(T) 22/225 (4/5) EE 78/278A: Random Variables Page 2 26
14 Mean and Variance of Famous Discrete RVs Bernoulli r.v. X Bern(p): The mean is The second moment is Thus the variance is E(X) p +( p) 0 p E(X 2 ) p 2 +( p) 0 2 p Var(X) E(X 2 ) (E(X)) 2 p p 2 p( p) Binomial r.v. X B(n,p): It is not easy to find it using the definition. Later we use a much simpler method to show that E(X) np Var(X) np( p) Just n times the mean (variance) of a Bernoulli! EE 78/278A: Random Variables Page 2 27 Geometric r.v. X Geom(p): The mean is E(X) kp( p) k k p p p p k( p) k k k( p) k k0 p p p p 2 p Note: We found the series sum using the following trick: Recall that for a <, the sum of the geometric series k0 a k a EE 78/278A: Random Variables Page 2 28
15 Suppose we want to evaluate ka k k0 Multiply both sides of the geometric series sum by a to obtain k0 a k+ a a Now differentiate both sides with respect to a. The LHS gives d da a k+ k0 (k +)a k a k0 ka k k0 The RHS gives Thus we have d a da a ( a) 2 ka k k0 a ( a) 2 EE 78/278A: Random Variables Page 2 29 The second moment can be similarly evaluated to obtain E(X 2 ) 2 p p 2 The variance is thus Var(X) E(X 2 ) (E(X)) 2 p p 2 EE 78/278A: Random Variables Page 2 30
16 Poisson r.v. X Poisson(λ): The mean is given by E(X) k0 e λ k λk k! e λ k e λ k λe λ k λe λ λ k0 Can show that the variance is also equal to λ k λk k! λ k (k )! λ (k ) (k )! λ k k! EE 78/278A: Random Variables Page 2 3 Mean and Variance for Continuous RVs Now consider a continuous r.v. X f X (), the epected value (or mean) of X is defined as E(X) f X ()d This has the interpretation of the center of mass for a mass density The second moment and variance are similarly defined as: E(X 2 ) 2 f X ()d Var(X) E [ (X E(X)) 2] E(X 2 ) (E(X)) 2 EE 78/278A: Random Variables Page 2 32
17 Uniform r.v. X U[a,b]: The mean, second moment, and variance are given by E(X) E(X 2 ) b a f X ()d 2 f X ()d 2 b a d b b 2 d b a a b a 3 b b3 a 3 3 a 3(b a) Var(X) E(X 2 ) (E(X)) 2 b3 a 3 3(b a) (a+b)2 (b a)2 4 2 Thus, for X U[0,], E(X) /2 and Var /2 a a+b d b a 2 EE 78/278A: Random Variables Page 2 33 Eponential r.v. X Ep(λ): The mean and variance are given by E(X) E(X 2 ) 0 f X ()d λe λ d ( e λ ) 0 0 λ e λ λ 2 λe λ d 2 λ 2 Var(X) 2 λ 2 λ 2 λ 2 e λ d (integration by parts) For a Gaussian r.v. X N(µ,σ 2 ), the mean is µ and the variance is σ 2 (will show this later using transforms) EE 78/278A: Random Variables Page 2 34
18 Mean and Variance for Famous r.v.s Random Variable Mean Variance Bern(p) p p( p) Geom(p) /p ( p)/p 2 B(n,p) np np( p) Poisson(λ) λ λ U[a,b] (a+b)/2 (b a) 2 /2 Ep(λ) /λ /λ 2 N(µ,σ 2 ) µ σ 2 EE 78/278A: Random Variables Page 2 35 Cumulative Distribution Function (cdf) For discrete r.v.s we use pmf s, for continuous r.v.s we use pdf s Many real-world r.v.s are mied, that is, have both discrete and continuous components Eample: A packet arrives at a router in a communication network. If the input buffer is empty (happens with probability p), the packet is serviced immediately. Otherwise the packet must wait for a random amount of time as characterized by a pdf Define the r.v. X to be the packet service time. X is neither discrete nor continuous There is a third probability function that characterizes all random variable types discrete, continuous, and mied. The cumulative distribution function or cdf F X () of a random variable is defined by F X () P{X } for (, ) EE 78/278A: Random Variables Page 2 36
19 Properties of the cdf: Like the pmf (but unlike the pdf), the cdf is the probability of something. Hence, 0 F X () Normalization aiom implies that F X ( ), and F X ( ) 0 F X () is monotonically nondecreasing, i.e., if a > b then F X (a) F X (b) The probability of any event can be easily computed from a cdf, e.g., for an interval (a, b], P{X (a,b]} P{a < X b} P{X b} P{X a} (additivity) F X (b) F X (a) The probability of any outcome a is: P{X a} F X (a) F X (a ) EE 78/278A: Random Variables Page 2 37 If a r.v. is discrete, its cdf consists of a set of steps p X () F X () If X is a continuous r.v. with pdf f X (), then F X () P{X } f X (α)dα So, the cdf of a continuous r.v. X is continuous F X () EE 78/278A: Random Variables Page 2 38
20 In fact the precise way to define a continuous random variable is through the continuity of its cdf Further, if F X () is differentiable (almost everywhere), then f X () df X() d F X (+ ) F X () lim 0 lim 0 P{ < X + } The cdf of a mied r.v. has the general form F X () EE 78/278A: Random Variables Page 2 39 Eamples cdf of a uniform r.v.: 0 if < a a F X () a b adα b a if a b if b f X () F X () b a a b a b EE 78/278A: Random Variables Page 2 40
21 cdf of an eponential r.v.: F X () 0, X < 0, and F X () e λ, 0 f X () F X () λ EE 78/278A: Random Variables Page 2 4 cdf for a mied r.v.: Let X be the service time of a packet at a router. If the buffer is empty (happens with probability p), the packet is serviced immediately. If it is not empty, service time is described by an eponential pdf with parameter λ > 0 The cdf of X is 0 if < 0 F X () p if 0 p+( p)( e λ ) if > 0 F X () p EE 78/278A: Random Variables Page 2 42
22 cdf of a Gaussian r.v.: There is no nice closed form for the cdf of a Gaussian r.v., but there are many published tables for the cdf of a standard normal pdf N(0,), the Φ function: Φ() 2π e ξ2 2 dξ More commonly, the tables are for the Q() Φ() function N(0,) Q() Or, for the complementary error function: erfc() 2Q( 2) for > 0 As we shall see, the Q( ) function can be used to quickly compute P{X > a} for any Gaussian r.v. X EE 78/278A: Random Variables Page 2 43 Functions of a Random Variable We are often given a r.v. X with a known distribution (pmf, cdf, or pdf), and some function y g() of, e.g., X 2, X, cosx, etc., and wish to specify Y, i.e., find its pmf, if it is discrete, pdf, if continuous, or cdf Each of these functions is a random variable defined over the original eperiment as Y(ω) g(x(ω)). However, since we do not assume knowledge of the sample space or the probability measure, we need to specify Y directly from the pmf, pdf, or cdf of X First assume that X p X (), i.e., a discrete random variable, then Y is also discrete and can be described by a pmf p Y (y). To find it we find the probability of the inverse image {ω : Y(ω) y} for every y. Assuming Ω is discrete: g( i ) y y y 2... y... EE 78/278A: Random Variables Page 2 44
23 Thus p Y (y) P{ω : Y(ω) y} P{ω} {ω: g(x(ω))y} {: g()y} p Y (y) {ω: X(ω)} {: g()y} P{ω} p X () {: g()y} p X () We can derive p Y (y) directly from p X () without going back to the original random eperiment EE 78/278A: Random Variables Page 2 45 Eample: Let the r.v. X be the maimum of two independent rolls of a four-sided die. Define a new random variable Y g(x), where g() { if 3 0 otherwise Find the pmf for Y Solution: p Y (y) p X () {: g()y} p Y () {: 3} p X () p Y (0) p Y () 4 EE 78/278A: Random Variables Page 2 46
24 Derived Densities Let X be a continuous r.v. with pdf f X () and Y g(x) such that Y is a continuous r.v. We wish to find f Y (y) Recall derived pmf approach: Given p X () and a function Y g(x), the pmf of Y is given by p Y (y) p X (), {: g()y} i.e., p Y (y) is the sum of p X () over all that yield g() y Idea does not etend immediately to deriving pdfs, since pdfs are not probabilities, and we cannot add probabilities of points But basic idea does etend to cdfs EE 78/278A: Random Variables Page 2 47 Can first calculate the cdf of Y as F Y (y) P{g(X) y} f X ()d {: g() y} y { : g() y} y Then differentiate to obtain the pdf f Y (y) df Y(y) dy The hard part is typically getting the limits on the integral correct. Often they are obvious, but sometimes they are more subtle EE 78/278A: Random Variables Page 2 48
25 Eample: Linear Functions Let X f X () and Y ax +b for some a > 0 and b. (The case a < 0 is left as an eercise) To find the pdf of Y, we use the above procedure y y y b a EE 78/278A: Random Variables Page 2 49 Thus F Y (y) P{Y y} P{aX +b y} { P X y b } ( ) y b F X a a f Y (y) df Y(y) dy Can show that for general a 0, Eample: X Ep(λ), i.e., Then f Y (y) f Y (y) a f X a f X ( ) y b a ( ) y b a f X () λe λ, 0 { λ a e λ(y b)/a if y b a 0 0 otherwise EE 78/278A: Random Variables Page 2 50
26 Eample: X N(µ,σ 2 ), i.e., Again setting Y ax +b, f X () f Y (y) a f X 2πσ 2 e ( µ)2 2σ 2 ( ) y b a a 2πσ 2 e ( y b a µ )2 2σ 2 2π(aσ) 2 e (y b aµ)2 2a 2 σ 2 e (y µ Y ) 2σ Y 2 2πσ 2 Y 2 for < y < Therefore, Y N(aµ+b,a 2 σ 2 ) EE 78/278A: Random Variables Page 2 5 This result can be used to compute probabilities for an arbitrary Gaussian r.v. from knowledge of the distribution a N(0,) r.v. Suppose Y N(µ Y,σY 2 ) and we wish to find P{Y > y} From the above result, we can epress Y σ Y X +µ Y, where X N(0,) Thus P{Y > y} P{σ Y X +µ Y > y} { P X > y µ } Y ( y µy Q σ Y σ Y ) EE 78/278A: Random Variables Page 2 52
27 Eample: A Nonlinear Function John is driving a distance of 80 miles at constant speed that is uniformly distributed between 30 and 60 miles/hr. What is the pdf of the duration of the trip? Solution: Let X be John s speed, then The duration of the trip is Y 80/X { /30 if f X () 0 otherwise, EE 78/278A: Random Variables Page 2 53 To find f Y (y) we first find F Y (y). Note that {y : Y y} { : X 80/y}, thus F Y (y) 80/y f X ()d 0 if y /y f X()d if 3 < y 6 if y > 6 0 if y 3 (2 6 y ) if 3 < y 6 if y > 6 Differentiating, we obtain { 6/y 2 if 3 y 6 f Y (y) 0 otherwise EE 78/278A: Random Variables Page 2 54
28 Monotonic Functions Let g() be a monotonically increasing and differentiable function over its range Then g is invertible, i.e., there eists a function h, such that y g() if and only if h(y) Often this is written as g. If a function g has these properties, then g() y iff h(y), and we can write F Y (y) P{g(X) y} P{X h(y)} F X (h(y)) h(y) f X ()d Thus f Y (y) df Y(y) dy f X (h(y)) dh dy (y) EE 78/278A: Random Variables Page 2 55 Generalizing the result to both monotonically increasing and decreasing functions yields f Y (y) f X (h(y)) dh dy (y) Eample: Recall the X U[30,60] eample with Y g(x) 80/X The inverse is X h(y) 80/Y Applying the above formula in region of interest Y [3,6] (it is 0 outside) yields f Y (y) f X (h(y)) dh dy (y) y 2 6 y 2 for 3 y 6 EE 78/278A: Random Variables Page 2 56
29 Another Eample: Suppose that X has a pdf that is nonzero only in [0,] and define Y g(x) X 2 In the region of interest the function is invertible and X Y Applying the pdf formula, we obtain f Y (y) f X (h(y)) dh dy (y) f X( y) 2 y, for 0 < y Personally I prefer the more fundamental approach, since I often forget this formula or mess up the signs EE 78/278A: Random Variables Page 2 57
Lecture Notes 2 Random Variables. Random Variable
Lecture Notes 2 Random Variables Definition Discrete Random Variables: Probability mass function (pmf) Continuous Random Variables: Probability density function (pdf) Mean and Variance Cumulative Distribution
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationLecture Notes 1 Probability and Random Variables. Conditional Probability and Independence. Functions of a Random Variable
Lecture Notes 1 Probability and Random Variables Probability Spaces Conditional Probability and Independence Random Variables Functions of a Random Variable Generation of a Random Variable Jointly Distributed
More informationComputer Science, Informatik 4 Communication and Distributed Systems. Simulation. Discrete-Event System Simulation. Dr.
Simulation Discrete-Event System Simulation Chapter 4 Statistical Models in Simulation Purpose & Overview The world the model-builder sees is probabilistic rather than deterministic. Some statistical model
More informationChapter 5. Statistical Models in Simulations 5.1. Prof. Dr. Mesut Güneş Ch. 5 Statistical Models in Simulations
Chapter 5 Statistical Models in Simulations 5.1 Contents Basic Probability Theory Concepts Discrete Distributions Continuous Distributions Poisson Process Empirical Distributions Useful Statistical Models
More informationChapter 2. Random Variable. Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance.
Chapter 2 Random Variable CLO2 Define single random variables in terms of their PDF and CDF, and calculate moments such as the mean and variance. 1 1. Introduction In Chapter 1, we introduced the concept
More informationRandom variables. DS GA 1002 Probability and Statistics for Data Science.
Random variables DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Motivation Random variables model numerical quantities
More informationLecture Notes 7 Random Processes. Markov Processes Markov Chains. Random Processes
Lecture Notes 7 Random Processes Definition IID Processes Bernoulli Process Binomial Counting Process Interarrival Time Process Markov Processes Markov Chains Classification of States Steady State Probabilities
More informationFundamental Tools - Probability Theory II
Fundamental Tools - Probability Theory II MSc Financial Mathematics The University of Warwick September 29, 2015 MSc Financial Mathematics Fundamental Tools - Probability Theory II 1 / 22 Measurable random
More informationn px p x (1 p) n x. p x n(n 1)... (n x + 1) x!
Lectures 3-4 jacques@ucsd.edu 7. Classical discrete distributions D. The Poisson Distribution. If a coin with heads probability p is flipped independently n times, then the number of heads is Bin(n, p)
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 14: Continuous random variables Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin www.cs.cmu.edu/
More informationChapter 3, 4 Random Variables ENCS Probability and Stochastic Processes. Concordia University
Chapter 3, 4 Random Variables ENCS6161 - Probability and Stochastic Processes Concordia University ENCS6161 p.1/47 The Notion of a Random Variable A random variable X is a function that assigns a real
More informationLecture 3. Discrete Random Variables
Math 408 - Mathematical Statistics Lecture 3. Discrete Random Variables January 23, 2013 Konstantin Zuev (USC) Math 408, Lecture 3 January 23, 2013 1 / 14 Agenda Random Variable: Motivation and Definition
More informationWe introduce methods that are useful in:
Instructor: Shengyu Zhang Content Derived Distributions Covariance and Correlation Conditional Expectation and Variance Revisited Transforms Sum of a Random Number of Independent Random Variables more
More informationRandom variable X is a mapping that maps each outcome s in the sample space to a unique real number x, x. X s. Real Line
Random Variable Random variable is a mapping that maps each outcome s in the sample space to a unique real number,. s s : outcome Sample Space Real Line Eamples Toss a coin. Define the random variable
More informationSTAT2201. Analysis of Engineering & Scientific Data. Unit 3
STAT2201 Analysis of Engineering & Scientific Data Unit 3 Slava Vaisman The University of Queensland School of Mathematics and Physics What we learned in Unit 2 (1) We defined a sample space of a random
More informationBrief Review of Probability
Maura Department of Economics and Finance Università Tor Vergata Outline 1 Distribution Functions Quantiles and Modes of a Distribution 2 Example 3 Example 4 Distributions Outline Distribution Functions
More informationDS-GA 1002 Lecture notes 2 Fall Random variables
DS-GA 12 Lecture notes 2 Fall 216 1 Introduction Random variables Random variables are a fundamental tool in probabilistic modeling. They allow us to model numerical quantities that are uncertain: the
More informationMA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems
MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
More informationEE/CpE 345. Modeling and Simulation. Fall Class 5 September 30, 2002
EE/CpE 345 Modeling and Simulation Class 5 September 30, 2002 Statistical Models in Simulation Real World phenomena of interest Sample phenomena select distribution Probabilistic, not deterministic Model
More informationRandom variable X is a mapping that maps each outcome s in the sample space to a unique real number x, < x <. ( ) X s. Real Line
Random Variable Random variable is a mapping that maps each outcome s in the sample space to a unique real number, <
More informationChapter 3 Single Random Variables and Probability Distributions (Part 1)
Chapter 3 Single Random Variables and Probability Distributions (Part 1) Contents What is a Random Variable? Probability Distribution Functions Cumulative Distribution Function Probability Density Function
More informationMath 180A. Lecture 16 Friday May 7 th. Expectation. Recall the three main probability density functions so far (1) Uniform (2) Exponential.
Math 8A Lecture 6 Friday May 7 th Epectation Recall the three main probability density functions so far () Uniform () Eponential (3) Power Law e, ( ), Math 8A Lecture 6 Friday May 7 th Epectation Eample
More informationRandom Variables. Definition: A random variable (r.v.) X on the probability space (Ω, F, P) is a mapping
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
More informationRandom Variables LECTURE DEFINITION
LECTURE 3 Random Variables 3. DEFINITION It is often convenient to represent the outcome of a random eperiment by a number. A random variable(r.v.) is such a representation. To be more precise, let (Ω,F,
More information1 Presessional Probability
1 Presessional Probability Probability theory is essential for the development of mathematical models in finance, because of the randomness nature of price fluctuations in the markets. This presessional
More information1 Random Variable: Topics
Note: Handouts DO NOT replace the book. In most cases, they only provide a guideline on topics and an intuitive feel. 1 Random Variable: Topics Chap 2, 2.1-2.4 and Chap 3, 3.1-3.3 What is a random variable?
More informationChapter 3: Random Variables 1
Chapter 3: Random Variables 1 Yunghsiang S. Han Graduate Institute of Communication Engineering, National Taipei University Taiwan E-mail: yshan@mail.ntpu.edu.tw 1 Modified from the lecture notes by Prof.
More informationWhy study probability? Set theory. ECE 6010 Lecture 1 Introduction; Review of Random Variables
ECE 6010 Lecture 1 Introduction; Review of Random Variables Readings from G&S: Chapter 1. Section 2.1, Section 2.3, Section 2.4, Section 3.1, Section 3.2, Section 3.5, Section 4.1, Section 4.2, Section
More informationLecture 2: Repetition of probability theory and statistics
Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:
More informationChapter 3: Random Variables 1
Chapter 3: Random Variables 1 Yunghsiang S. Han Graduate Institute of Communication Engineering, National Taipei University Taiwan E-mail: yshan@mail.ntpu.edu.tw 1 Modified from the lecture notes by Prof.
More informationChapter 2: The Random Variable
Chapter : The Random Variable The outcome of a random eperiment need not be a number, for eample tossing a coin or selecting a color ball from a bo. However we are usually interested not in the outcome
More informationStatistics and Econometrics I
Statistics and Econometrics I Random Variables Shiu-Sheng Chen Department of Economics National Taiwan University October 5, 2016 Shiu-Sheng Chen (NTU Econ) Statistics and Econometrics I October 5, 2016
More informationAppendix A : Introduction to Probability and stochastic processes
A-1 Mathematical methods in communication July 5th, 2009 Appendix A : Introduction to Probability and stochastic processes Lecturer: Haim Permuter Scribe: Shai Shapira and Uri Livnat The probability of
More informationDiscrete Random Variables
Chapter 5 Discrete Random Variables Suppose that an experiment and a sample space are given. A random variable is a real-valued function of the outcome of the experiment. In other words, the random variable
More informationDiscrete Random Variables
CPSC 53 Systems Modeling and Simulation Discrete Random Variables Dr. Anirban Mahanti Department of Computer Science University of Calgary mahanti@cpsc.ucalgary.ca Random Variables A random variable is
More informationMidterm Exam 1 Solution
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2015 Kannan Ramchandran September 22, 2015 Midterm Exam 1 Solution Last name First name SID Name of student on your left:
More informationLecture 3: Random variables, distributions, and transformations
Lecture 3: Random variables, distributions, and transformations Definition 1.4.1. A random variable X is a function from S into a subset of R such that for any Borel set B R {X B} = {ω S : X(ω) B} is an
More informationRandom Variables Example:
Random Variables Example: We roll a fair die 6 times. Suppose we are interested in the number of 5 s in the 6 rolls. Let X = number of 5 s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the
More informationEE 302 Division 1. Homework 6 Solutions.
EE 3 Division. Homework 6 Solutions. Problem. A random variable X has probability density { C f X () e λ,,, otherwise, where λ is a positive real number. Find (a) The constant C. Solution. Because of the
More informationSystem Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models
System Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models Fatih Cavdur fatihcavdur@uludag.edu.tr March 29, 2014 Introduction Introduction The world of the model-builder
More informationReview of Probability. CS1538: Introduction to Simulations
Review of Probability CS1538: Introduction to Simulations Probability and Statistics in Simulation Why do we need probability and statistics in simulation? Needed to validate the simulation model Needed
More informationNorthwestern University Department of Electrical Engineering and Computer Science
Northwestern University Department of Electrical Engineering and Computer Science EECS 454: Modeling and Analysis of Communication Networks Spring 2008 Probability Review As discussed in Lecture 1, probability
More informationChapter 2: Random Variables
ECE54: Stochastic Signals and Systems Fall 28 Lecture 2 - September 3, 28 Dr. Salim El Rouayheb Scribe: Peiwen Tian, Lu Liu, Ghadir Ayache Chapter 2: Random Variables Example. Tossing a fair coin twice:
More informationLecture 2: CDF and EDF
STAT 425: Introduction to Nonparametric Statistics Winter 2018 Instructor: Yen-Chi Chen Lecture 2: CDF and EDF 2.1 CDF: Cumulative Distribution Function For a random variable X, its CDF F () contains all
More informationECS /1 Part IV.2 Dr.Prapun
Sirindhorn International Institute of Technology Thammasat University School of Information, Computer and Communication Technology ECS35 4/ Part IV. Dr.Prapun.4 Families of Continuous Random Variables
More informationPart IA Probability. Definitions. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Definitions Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationMathematical Statistics 1 Math A 6330
Mathematical Statistics 1 Math A 6330 Chapter 3 Common Families of Distributions Mohamed I. Riffi Department of Mathematics Islamic University of Gaza September 28, 2015 Outline 1 Subjects of Lecture 04
More informationChapter 5. Random Variables (Continuous Case) 5.1 Basic definitions
Chapter 5 andom Variables (Continuous Case) So far, we have purposely limited our consideration to random variables whose ranges are countable, or discrete. The reason for that is that distributions on
More informationChapter 1 Statistical Reasoning Why statistics? Section 1.1 Basics of Probability Theory
Chapter 1 Statistical Reasoning Why statistics? Uncertainty of nature (weather, earth movement, etc. ) Uncertainty in observation/sampling/measurement Variability of human operation/error imperfection
More informationChapter 1: Revie of Calculus and Probability
Chapter 1: Revie of Calculus and Probability Refer to Text Book: Operations Research: Applications and Algorithms By Wayne L. Winston,Ch. 12 Operations Research: An Introduction By Hamdi Taha, Ch. 12 OR441-Dr.Khalid
More informationIntroduction to Probability Theory for Graduate Economics Fall 2008
Introduction to Probability Theory for Graduate Economics Fall 008 Yiğit Sağlam October 10, 008 CHAPTER - RANDOM VARIABLES AND EXPECTATION 1 1 Random Variables A random variable (RV) is a real-valued function
More informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Tobias Neckel, Ionuț-Gabriel Farcaș Lehrstuhl Informatik V Summer Semester 2017 Lecture 2: Repetition of probability theory and statistics Example: coin flip Example
More informationTopic 3: The Expectation of a Random Variable
Topic 3: The Expectation of a Random Variable Course 003, 2017 Page 0 Expectation of a discrete random variable Definition (Expectation of a discrete r.v.): The expected value (also called the expectation
More information1: PROBABILITY REVIEW
1: PROBABILITY REVIEW Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) Slides 1: Probability Review 1 / 56 Outline We will review the following
More informationLecture 1: August 28
36-705: Intermediate Statistics Fall 2017 Lecturer: Siva Balakrishnan Lecture 1: August 28 Our broad goal for the first few lectures is to try to understand the behaviour of sums of independent random
More information3 Multiple Discrete Random Variables
3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f
More informationClass 26: review for final exam 18.05, Spring 2014
Probability Class 26: review for final eam 8.05, Spring 204 Counting Sets Inclusion-eclusion principle Rule of product (multiplication rule) Permutation and combinations Basics Outcome, sample space, event
More information18.440: Lecture 28 Lectures Review
18.440: Lecture 28 Lectures 18-27 Review Scott Sheffield MIT Outline Outline It s the coins, stupid Much of what we have done in this course can be motivated by the i.i.d. sequence X i where each X i is
More informationProbability, Random Processes and Inference
INSTITUTO POLITÉCNICO NACIONAL CENTRO DE INVESTIGACION EN COMPUTACION Laboratorio de Ciberseguridad Probability, Random Processes and Inference Dr. Ponciano Jorge Escamilla Ambrosio pescamilla@cic.ipn.mx
More informationS n = x + X 1 + X X n.
0 Lecture 0 0. Gambler Ruin Problem Let X be a payoff if a coin toss game such that P(X = ) = P(X = ) = /2. Suppose you start with x dollars and play the game n times. Let X,X 2,...,X n be payoffs in each
More informationCMPSCI 240: Reasoning Under Uncertainty
CMPSCI 240: Reasoning Under Uncertainty Lecture 5 Prof. Hanna Wallach wallach@cs.umass.edu February 7, 2012 Reminders Pick up a copy of B&T Check the course website: http://www.cs.umass.edu/ ~wallach/courses/s12/cmpsci240/
More informationELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 2 Discrete Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Random Variable Discrete Random
More informationPCMI Introduction to Random Matrix Theory Handout # REVIEW OF PROBABILITY THEORY. Chapter 1 - Events and Their Probabilities
PCMI 207 - Introduction to Random Matrix Theory Handout #2 06.27.207 REVIEW OF PROBABILITY THEORY Chapter - Events and Their Probabilities.. Events as Sets Definition (σ-field). A collection F of subsets
More informationChapter 6 Expectation and Conditional Expectation. Lectures Definition 6.1. Two random variables defined on a probability space are said to be
Chapter 6 Expectation and Conditional Expectation Lectures 24-30 In this chapter, we introduce expected value or the mean of a random variable. First we define expectation for discrete random variables
More informationProbability Theory Review
Cogsci 118A: Natural Computation I Lecture 2 (01/07/10) Lecturer: Angela Yu Probability Theory Review Scribe: Joseph Schilz Lecture Summary 1. Set theory: terms and operators In this section, we provide
More informationSTAT 430/510 Probability Lecture 12: Central Limit Theorem and Exponential Distribution
STAT 430/510 Probability Lecture 12: Central Limit Theorem and Exponential Distribution Pengyuan (Penelope) Wang June 15, 2011 Review Discussed Uniform Distribution and Normal Distribution Normal Approximation
More informationThings to remember when learning probability distributions:
SPECIAL DISTRIBUTIONS Some distributions are special because they are useful They include: Poisson, exponential, Normal (Gaussian), Gamma, geometric, negative binomial, Binomial and hypergeometric distributions
More informationStatistics for Economists. Lectures 3 & 4
Statistics for Economists Lectures 3 & 4 Asrat Temesgen Stockholm University 1 CHAPTER 2- Discrete Distributions 2.1. Random variables of the Discrete Type Definition 2.1.1: Given a random experiment with
More informationContinuous random variables
Continuous random variables Continuous r.v. s take an uncountably infinite number of possible values. Examples: Heights of people Weights of apples Diameters of bolts Life lengths of light-bulbs We cannot
More informationDiscrete Random Variable
Discrete Random Variable Outcome of a random experiment need not to be a number. We are generally interested in some measurement or numerical attribute of the outcome, rather than the outcome itself. n
More informationRelationship between probability set function and random variable - 2 -
2.0 Random Variables A rat is selected at random from a cage and its sex is determined. The set of possible outcomes is female and male. Thus outcome space is S = {female, male} = {F, M}. If we let X be
More informationRandom Variables. Random variables. A numerically valued map X of an outcome ω from a sample space Ω to the real line R
In probabilistic models, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. As a function or a map, it maps from an element (or an outcome) of a sample
More informationStochastic Models in Computer Science A Tutorial
Stochastic Models in Computer Science A Tutorial Dr. Snehanshu Saha Department of Computer Science PESIT BSC, Bengaluru WCI 2015 - August 10 to August 13 1 Introduction 2 Random Variable 3 Introduction
More informationThe random variable 1
The random variable 1 Contents 1. Definition 2. Distribution and density function 3. Specific random variables 4. Functions of one random variable 5. Mean and variance 2 The random variable A random variable
More informationAnalysis of Engineering and Scientific Data. Semester
Analysis of Engineering and Scientific Data Semester 1 2019 Sabrina Streipert s.streipert@uq.edu.au Example: Draw a random number from the interval of real numbers [1, 3]. Let X represent the number. Each
More informationRecap of Basic Probability Theory
02407 Stochastic Processes Recap of Basic Probability Theory Uffe Høgsbro Thygesen Informatics and Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark Email: uht@imm.dtu.dk
More informationProbability Review. Yutian Li. January 18, Stanford University. Yutian Li (Stanford University) Probability Review January 18, / 27
Probability Review Yutian Li Stanford University January 18, 2018 Yutian Li (Stanford University) Probability Review January 18, 2018 1 / 27 Outline 1 Elements of probability 2 Random variables 3 Multiple
More informationChapter 3 Discrete Random Variables
MICHIGAN STATE UNIVERSITY STT 351 SECTION 2 FALL 2008 LECTURE NOTES Chapter 3 Discrete Random Variables Nao Mimoto Contents 1 Random Variables 2 2 Probability Distributions for Discrete Variables 3 3 Expected
More informationSpecial distributions
Special distributions August 22, 2017 STAT 101 Class 4 Slide 1 Outline of Topics 1 Motivation 2 Bernoulli and binomial 3 Poisson 4 Uniform 5 Exponential 6 Normal STAT 101 Class 4 Slide 2 What distributions
More informationRecitation 2: Probability
Recitation 2: Probability Colin White, Kenny Marino January 23, 2018 Outline Facts about sets Definitions and facts about probability Random Variables and Joint Distributions Characteristics of distributions
More informationQuick Tour of Basic Probability Theory and Linear Algebra
Quick Tour of and Linear Algebra Quick Tour of and Linear Algebra CS224w: Social and Information Network Analysis Fall 2011 Quick Tour of and Linear Algebra Quick Tour of and Linear Algebra Outline Definitions
More informationLecture 16. Lectures 1-15 Review
18.440: Lecture 16 Lectures 1-15 Review Scott Sheffield MIT 1 Outline Counting tricks and basic principles of probability Discrete random variables 2 Outline Counting tricks and basic principles of probability
More informationRecap of Basic Probability Theory
02407 Stochastic Processes? Recap of Basic Probability Theory Uffe Høgsbro Thygesen Informatics and Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark Email: uht@imm.dtu.dk
More informationCSE 312, 2017 Winter, W.L. Ruzzo. 7. continuous random variables
CSE 312, 2017 Winter, W.L. Ruzzo 7. continuous random variables The new bit continuous random variables Discrete random variable: values in a finite or countable set, e.g. X {1,2,..., 6} with equal probability
More informationSTAT Chapter 5 Continuous Distributions
STAT 270 - Chapter 5 Continuous Distributions June 27, 2012 Shirin Golchi () STAT270 June 27, 2012 1 / 59 Continuous rv s Definition: X is a continuous rv if it takes values in an interval, i.e., range
More information1 Review of Probability
1 Review of Probability Random variables are denoted by X, Y, Z, etc. The cumulative distribution function (c.d.f.) of a random variable X is denoted by F (x) = P (X x), < x
More informationContinuous Random Variables
Continuous Random Variables Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability. Often, there is interest in random variables
More informationNotes 6 Autumn Example (One die: part 1) One fair six-sided die is thrown. X is the number showing.
MAS 08 Probability I Notes Autumn 005 Random variables A probability space is a sample space S together with a probability function P which satisfies Kolmogorov s aioms. The Holy Roman Empire was, in the
More informationMore on Distribution Function
More on Distribution Function The distribution of a random variable X can be determined directly from its cumulative distribution function F X. Theorem: Let X be any random variable, with cumulative distribution
More information2. Suppose (X, Y ) is a pair of random variables uniformly distributed over the triangle with vertices (0, 0), (2, 0), (2, 1).
Name M362K Final Exam Instructions: Show all of your work. You do not have to simplify your answers. No calculators allowed. There is a table of formulae on the last page. 1. Suppose X 1,..., X 1 are independent
More informationRS Chapter 2 Random Variables 9/28/2017. Chapter 2. Random Variables
RS Chapter Random Variables 9/8/017 Chapter Random Variables Random Variables A random variable is a convenient way to epress the elements of Ω as numbers rather than abstract elements of sets. Definition:
More informationContinuous Probability Spaces
Continuous Probability Spaces Ω is not countable. Outcomes can be any real number or part of an interval of R, e.g. heights, weights and lifetimes. Can not assign probabilities to each outcome and add
More informationNotes for Math 324, Part 20
7 Notes for Math 34, Part Chapter Conditional epectations, variances, etc.. Conditional probability Given two events, the conditional probability of A given B is defined by P[A B] = P[A B]. P[B] P[A B]
More informationStochastic processes Lecture 1: Multiple Random Variables Ch. 5
Stochastic processes Lecture : Multiple Random Variables Ch. 5 Dr. Ir. Richard C. Hendriks 26/04/8 Delft University of Technology Challenge the future Organization Plenary Lectures Book: R.D. Yates and
More information1.1 Review of Probability Theory
1.1 Review of Probability Theory Angela Peace Biomathemtics II MATH 5355 Spring 2017 Lecture notes follow: Allen, Linda JS. An introduction to stochastic processes with applications to biology. CRC Press,
More informationWeek 12-13: Discrete Probability
Week 12-13: Discrete Probability November 21, 2018 1 Probability Space There are many problems about chances or possibilities, called probability in mathematics. When we roll two dice there are possible
More informationProbability Theory and Random Variables
Probability Theory and Random Variables One of the most noticeable aspects of many computer science related phenomena is the lack of certainty. When a job is submitted to a batch oriented computer system,
More informationBrief Review of Probability
Brief Review of Probability Nuno Vasconcelos (Ken Kreutz-Delgado) ECE Department, UCSD Probability Probability theory is a mathematical language to deal with processes or experiments that are non-deterministic
More informationLecture Notes 1 Basic Probability. Elements of Probability. Conditional probability. Sequential Calculation of Probability
Lecture Notes 1 Basic Probability Set Theory Elements of Probability Conditional probability Sequential Calculation of Probability Total Probability and Bayes Rule Independence Counting EE 178/278A: Basic
More information