Topic 5: Discrete Random Variables & Expectations Reference Chapter 4
|
|
- Charlene Clarke
- 5 years ago
- Views:
Transcription
1 Page 1 Topic 5: Discrete Random Variables & Epectations Reference Chapter 4 In Chapter 3 we studied rules for associating a probability value with a single event or with a subset of events in an eperiment. Now we will epand our analysis to consider all possible events in an eperiment. A random variable is a which assigns values to each possible outcome of an eperiment.
2 Page 2 Note: May need to distinguish between the of a random variable, and its value. Eample:{X=} means the variable X takes the value. Eample: Eperiment: choose student; ask if he/she is planning to major in business. Two discrete outcomes: Yes; No. Define random variables: = = 1; 0; Yes No = 0, 1 } indicator variable
3 Page 3 Eample: Midterm Test. Students may get a grade A to F Define: X=5; A X=4; B X=3; C X=2; D X=1; F Depending on the contet, sample space may be or continuous. If discrete, it may be finite or (countably).
4 Probability Distributions Page 4 Once an eperiment and its outcomes have been clearly defined and the random variable of interest has been defined, then the probability of the occurrence of any of the random variable can be specified. I.e. Can now specify the probability of each value of the random variable occurring.
5 Page 5 Eample: There are 5 projects to be done. Twenty-five government workers are to be allocated to teams (varying size) to work on them: Outcome Project # of Employees Value of X () Prob(X=) 1 1 5/ / / / / Probability Distribution
6 Can also depict graphically: Page 6 P(X) X The table or graph represent the probability distribution of X: Each value of X and its of occurrence. Similar to our earlier notion of frequency distribution, but now relates to all possible situations that can occur.
7 We can use probability distribution to determine probabilities of interesting events: Page 7 For Eample: P( 2) = P( assigned to project 2 or 3 or 4 or 5) = P( = 2) + P( = 3) + P( = 4) + P( = 5) = = =
8 Definitions: Page 8 If the individual values assigned to the random variable can be, it is called a discrete random variable. In this case, its probability distribution is called a probability mass function,. Note: Defining Characteristics of P.M.F.: () i 0 P( X = ) 1 ( ii) P( X = ) = 1
9 In the same manner, we can define the cumulative function (c.m.f.): Page 9 * * F( X ) = P( X ) = P( ) * The value of c.m.f. at any point is usually denoted F(*), where it is the of all values of the p.m.f. for all the values of the random variable that are *.
10 Page 10 Eample: Long-term eperience suggests the following p.m.f. for sales of flu medication in January: Type X # P(X) F(X) Aspirin for Flu Tylenol Flu Benadryl F(2) =P(=1 or =2)=P(Aspirin or Tylenol) If a customer buys flu medicine in January, the probability it is not Benadryl is 0.60.
11 P(>1)=1-P( 1)=1-P(=1)= =0.9 Page 11 P(1< 3)=P(=3)+P(=2)=0.9 =F(3)-F(1)=1-0.10=0.9 P(2< 3)=P(=3)=0.4 =F(3)-F(2)=1-0.60=0.4
12 Page 12 When we previously considered data distributions, we looked at summary measures such as the mean, variance, etc.. A distribution is just a special type of data distribution same motivation here to evaluate such measures. To do this, we define: The E Value: of the random variable X is: a weighted average of all values with μ = E( ) = P( ) probabilities as weights.
13 Page 13 The epected value of a random variable X is found by multiplying each value of the random variable by its probability and then summing all these products. The letter E usually denotes an e value. The epected value of X is the balancing point for the p.m.f..
14 Eample: Marks obtained by students: Marks X P() P() (midpoint) 0 < < < < < µ=e(x)=50 Page 14
15 Page 15 Sometimes we want to work out the epected value of some f of a random variable. For instance, instead of finding the mean of the random variable X, we might be interested in determining the epected of X 2, or Log X, or of e. If X is a random variable, then these functions of X are also variables, and their epected values can be determined. The epected value of g(): Eg ( ) = gp ( ) ( ) [ ] all
16 Eamples: Page 16 [ ] 2 2 E = ( ) P ( ) [ ] E log( ) = log( ) P( ) [ ] E ( + 2) = ( + 2) P( ) (Watch the units.)
17 Eample: Epected compounded return on stocks over 3 years: Stock 1-Yr. P() 3 P() Return () A B C E( 3 )= Page 17 Epected return over 3 years is 38.97%.
18 The epectation operator, E( ), can be used to help us define the v of a random variable. Page 18 Recall, the variance of a frequency distribution was: 1 N ( ) 2 μ f i i (Average of ( i -µ) 2 values.) So, the of a random variable, X, is: [ ] V( ) = σ = ( μ) P( ) = P( ) μ
19 Eample: Volatility of Stock returns over one year: µ =E()=11.5% Page 19 Stock Return () μ 2 A 15% B 10% C 5% P() ( ) P( )
20 Rules of Epectation : E ( ) = P ( ) Page 20 1) The E(c)=c, where c is a constant. E( c) = cp( c) = c 1 = c eg. E(10)= 2) V(c)=0; The variance of a constant is zero. Recall: V()=E(-µ) 2 ; V(c)=E(c-µ) 2 =E(c-c) 2 =E(0) 2 =0 C X
21 3) E(aX)=aE(X): Multiply X by a constant: = = = E( a) ap( ) a P( ) ae( ) eg. E(10)=10E() Page 21 4) E(+b)=E()+b: Add a constant E( + b) = ( + b) P( ) = ( b) P( ) + ( b) P( ) = E( ) + b P( ) = E( ) + b since P( ) = 1.
22 5) V(a)=a 2 V(): Recall V()=E(-µ) 2 ; Page 22 V(a)=E(a-E(a)) 2 =E(a-aE()) 2 =E(a(-E())) 2 =E(a 2 (-E()) 2 ) V(a)=a 2 E(-E()) 2 =a 2 V() Eample: V(120)=
23 6) V(+b)=V() Page 23 V(+b)=E((+b)-E(+b)) 2 =E(+b-E()-b) 2 =E(-E()) 2 =V() E(a±b)=aE() ±b V(a±b)= Eamples: V(+10)= V(-20)=V() ***************************
24 One advantage of knowing these rules is to speed up calculations. Page 24 Eample: Data in U.S. $: epected value=u.s. $10 and standard deviation =U.S. $2. What about the results in $ CAN? Suppose C$1 =U.S. $.996 (U.S. $1 =C$1.004) E(a)=aE(); a =1.004 So, epected value C$ ( )= C$10.04 V(a)= a 2 V(); a= Std. deviation (a)= a S.D. (X) =C$( )=C$2.008
25 Eample: F=random variable of temperature (ºF) E(F)= 72.4ºF; V(F)=64(ºF) 2. Page 25 What is E(C) and V(C), where C is temperature (ºC)? F = C; C = F 32 = F 9 5 And since E(a+b)=aE()+b: E( C) = E( F) = b 2 V( C) = ( ) V( F) = a a
26 Page 26 Note: these epectation and variance rules also enable us to standardize random variables for comparison purposes: E()= μ ; V()=σ 2 ; S.D.()=σ. X μ Z = X Then let: = 1 { μ σ σ { σ 1 EZ ( ) = EX ( ) μ σ σ = 1 μ = σ μ σ 0 a So: E(Z)=0; V(Z)=1 b V( Z) = 2 V( X) 1 σ = = = 2 2 σ σ σ σ 1
27 Bivariate Probability Distributions: Page 27 In many situations, an eperiment may involve outcomes that are related to or more random variables This section considers probability functions that involve more than one variable: such functions are called multivariate probability functions We will only deal with the case of bivariate (two variables) probability functions. When a sample space involves two or more random variables, the function describing their combined probability is called a j probability function.
28 Eample: Behaviour of Households: Page 28 X=Income of the oldest member of the household Y= Years work eperience of this member By looking at P(X= and Y=y) for all, y values, we get the JOINT PROBABILITY DISTRIBUTION. X Income ($) Eperience 25,000 30,000 50,000 (Years) Y Entries in the table are j probabilities!
29 Two properties for Joint Probability Functions: Page 29 () i 0 P(, y) 1 ( ii) P(, y) where P(,y)=P(X= and Y=y). y = 1 This joint distribution also helps us with calculation of conditional probabilities.
30 Page 30 X Income ($) Eperience 25,000 30,000 50,000 P(y) (Years) Y P() P( ) = P( X = ) = P( X = and Y = y) y = "marginal probability of X" Py ( ) = PY ( = y) = P( X = and Y = y) = "marginal probability of Y"
31 Page 31 A marginal probability for any value of Y may be found by summing all the probabilities involving that value of Y. The table depicts joint probability distributions of X and Y.
32 When we discussed events, we had: Page 32 P(A I B) P(A B) =. P(B) Conditional Joint Marginal or P(X= Y=y) = P(X = and Y = y) P(Y = y). P( y) = P(,y) P(y).
33 From the last eample: Page 33 (i) P($25,000 and 5 years eperience)= (ii) P($25,000)= ; P(5Years)=0.26 (iii) P($25,000 5 years eperience)= =
34 Independence Page 34 As with events, two random variables are in if: P( y) = P() for all and y. P(,y)=P()P(y); for all and y If the knowledge of a certain on one variable does not affect the probability of the occurrence of values of the other random variable, then the two variables are independent. I.e. X and Y are independent if the conditional probability of X given Y is the same as the marginal probability of X.
35 If X and Y are independent, the P(,y)=. Page 35 An eample of not independent from the last eample: P(=$25,000 and y=5 years)= P(=$25,000)= P(y=5 years) =0.26 However: (0.35)(0.26) Years of eperience and income are dependent random variables.
36 Epectations of Combined Random Variables: Page 36 When dealing with the joint probability distribution of X and Y, the rule for finding epectations is similar to the rule for finding the epected value of a function. Basic rule: [ ] Eg ( ) = gp ( ) ( ) all If we combine two random variables together into h(x,y), then: [ ] = EhXY (, ) hypy (, ) (, ) y
37 Eample: Selling items of different value: EXY [ ] = ( ypy ) (, ) y X Quantity Price P(y) ($) Y P() EXY [ ] = ypy ( ) (, ) y = [( 1 4 0) + ( ) + ( ) + ( ) + ( ) + ( ) + Page 37 ( ) + ( ) + ( ) = $
38 Covariance of X and Y Page 38 When we have 2 random variables it is interesting to measure the etent to which their random behaviours are related. The covariance, C(X,Y), measures how much the two random variables with each other, (how they co-vary). Look at the variation of each random variable relative to its respective. Let μ = μ = y E( X ) EY ( ): ( μ )( μ ) y [ ] [( μ )( μ )] y Cov( X, Y) = E y = y y P(, y) Cov( X, Y) = E( X, Y) E( X ) E( Y)
39 Note: if X and Y are independent: P(X,Y)=P(X) * P(Y) Page 39 [( )( )] μ μy Cov( X, Y) = y P(, y) = = y P( ) y P( y) y ( μ ) ( ) μy P ( ) P ( ) ypy ( ) μy Py ( ) y y ( ) μ ( ) [ μ ] [ ] μy = E() E(y) - = 0 = ( μ μ = 0)( μ μ = 0) y y
40 Page 40 If high values of X (relative to μ ) tend to be associated with high values of Y (relative to μ y ) then C(X,Y) will be a large number. If low values of one variable tend to be associated with high values of the other, then C(X,Y) will be a large number. Eample continued: X Quantity Price P(y) ($) Y P()
41 [ ] E Quantity( X ) = X P( X ) = [( ) + ( ) + ( ) = Page 41 [ ] E Price( Y) = Price P( price) y = [( ) + ( ) + ( ) = Cov[ X, Y] = ( μ)( y μy) P(, y) Y = [( )( )( 0)] + [( )( )( 0. 05) + L+ [( )( )( 0. 3)] = or: COV(, y) = E( XY) E( X ) E( Y) = [ (2.28)(5.55) = 0.016
42 Page 42 One limitation of the covariance measure is that it depends on u of the data, and it can take any positive or negative value. To compensate for both of these failings, especially when wanting to across different data sets, consider the following measure: Cov X Y ρ = y.(, ) = σσ y where σ σ correlation is the standard deviation of X, = variance() = E( μ ) 2
43 We call ρ y the simple correlation between the random variables X and Y. Page 43 () i ( ii) correlation is unitless. 1 ρ 1 X Quantity Price P(y) ($) Y P()
44 Page 44 [ ] E Quantity( X ) = X P( X ) = 228. [ Price ] E ( Y) = Y P( Y) = 555. y COV(, y) = E( XY) E( X) E( Y) =0.016 EXY [ ] = ( ypy ) (, ) = $ y
45 [ ] 2 2 E Quantity( X ) = X P( X ) = [( ) + ( ) + ( ) = 5. 7 Page 45 [ 2 ] E Price( Y ) = Y 2 P( Y) y = [( ) + ( ) + ( ) = VAR( ) = E( X ) [ E( )] σ = = [2.28] 2 =
46 VAR( Y) = E( y ) [ E( y)] 2 2 = [5.55] 2 = Page 46 σ y = ρ, y COV(, y) = = σσ y = = positive correlation between and y.
47 Linear Combinations of Random Variables Page 47 In general, we had: EhXY [ (, )] = hypy (, ) (, ) y Now consider the epectation of the weighted sum of: h(x,y)=ax +by where a and b are constants:
48 i) E(aX + by) = ( ax + by) P(,y) = ( ax) P(,y) + (by) P(,y) = a y [ ] P(,y) + b y P(,y) y y = a P() + b yp(y) = ae(x) + be(y) y y Page 48
49 More directly: E(aX +by) =E(aX) +E(bY) =a E(X) + b E(Y) (ii) Var(aX + by) [( ) ( )] = E ax + by E ax + by [( )] = E ax ae(x) + by be(y) = Ea(X E(X)) + b(y E(Y)) [ ] [ 2 2] [ 2 2] = E a (X E(X)) + E b (Y E(Y)) + E 2ab(X E(X))(Y E(Y)) [ ] 2 2 = a V(X) + b V(Y) + 2abCov.(X,Y) Page 49
50 Page 50 X Quantity Price P(y) ($) Y P() E[ Quantity( X )] = 228. E [ Y ] Price( ) = 555. COV(, y) =0.016 EXY [ ] = $
51 Page 51 [ 2 ] [ 2 Y ] = E Quantity( X ) = 57. E Price( ) VAR( ) = σ = VAR( Y) = σ = y ρ, y =
52 Page 52 Let R=3X+7Y. What is the epected value and the standard deviation of R? E(R)=3E() + 7E(y) 3(2.28)+7(5.55)=45.69 V(R)=V(3+7y) =9V()+49V(y)+2(3)(7)Cov(,y) =(9)(0.5016)+(49)(0.3475)+(2)(3)(7)(0.016) = = Std. Dev.=
53 Page 53
54 Discrete Probability Distributions Page 54 While it is often useful to determine probabilities for a specific discrete random variable or combined random variables, there are many situations in statistical inference and decision-making that involve the s type of probability function. I.e. Certain probability distributions have general characteristics which can be eploited. Recognize the similarities between certain types of eperiments and match a given case to the general formulas for mean, variance, independence and other characteristics of the random variables.
55 Page 55 The eamples we have seen so far have been different, specific, probability distributions. We are given information in each case. Let s consider one specific, common, discrete random variable. In Topic 6 we will consider one specific continuous random variable.
56 The Binomial Distribution: Page 56 Many eperiments share the common trait that their can be classified into one of two events. For Eample: 1. Select a person: male or 2. Apply for a job: Success or 3. Your credit card bill: paid or due It is often possible to describe the outcomes of events as either a success or failure. Use general terminology of success or failure to distinguish.
57 Page 57 Eperiments involving repeated independent trials, each with just two possible outcomes, form the basis of the distribution ( probability distribution). Definition: Bernoulli Trial: Each repetition of an eperiment involving only outcomes is called a Bernoulli trial. Interested in a series of independent, repeated Bernoulli trials. Independent results mean any one trial cannot influence the results of any other trial.
58 Note: Name the trials: t 1 t 2 t 3..., then,: P( t 2 =success t 1 = success)=p( t 2 =success); etc. Page 58 By repeated trials we mean that each time the eperiment is conducted, the conditions are. The Binomial Distribution is characterized by: n = number of Π= probability of success in one independent trial. The values of n and Π are referred to as the parameters of this distribution. They help determine the of success and failure.
59 Page 59 In a binomial distribution, the probabilities of interest are those of receiving a certain number () of in n independent, each trial having the same probability (Π) of success. Note: P(Failure) = 1- P(Success) as there are only 2 possible outcomes. They are complementary outcomes.
60 The Binomial Formula: Page 60 To determine the probability of eactly successes in n repeated Bernoulli trials, each with a probability of success equal to Π, it is necessary to find the probability of any one of outcomes where there are successes. If there are successes in n trials, there must be (n-) failures. The probabilities of interest are those of getting successes in n independent trials, each of which has probability Π of success. P(Event)=P(Outcome #1) + P(Outcome #2)+... = P(Outcomes) * (# of Outcomes) if each outcome has the same probability.
61 We can choose from n in : n! =!(n ) ways. Page 61 So, there are nc outcomes consistent with the event of s and (n-) failures. Each of these (nc) outcomes occurs with probability n [ Π ( 1 Π) ] So:.
62 P(X successes in n trials) n = Π ( 1 Π) ( nc) n! n = n Π ( 1 Π)!( )! = 0, 1, 2, 3,...,n for n = 1, 2, 3,... etc. Page 62 This set of probabilities gives the probability distribution for a Random Variable.
63 Note: If is B(n, Π), it is a random variable: Page 63 P( X) = nc( Π) ( 1 Π) n = 0, 1, 2, 3,...,n n = 1, 2, 3,... is its probability function: So: () i P( X) 0; all. n = 0 ( ii) P( X ) = 1
64 Eample: Π P( 0) = C ( 02. ) ( 08. ) 4 0 = 02. ; n = 4 ; ( 1 Π) = ! 4 = ( 1)( 08. ) = 0. 0!4! Page 64 P( 1) = C( 02. ) ( 08. ) ! 1 3 = ( 02. ) ( 08. ) = 0. 1!3! P( 2) = C ( 02. ) ( 08. ) = 4! (.)(.) = 2!2!
65 P() 3 = C (.)(.) = 4! (.)(.) = 3!1! Page 65 P( 4) = C ( 02. ) ( 08. ) 4 4 = ! 4 0 ( 02. ) ( 08. ) = 0. 4!0! P(X) X
66 Binomial tables assist in the calculations. Page 66 General Characteristics of Binomial Distribution (A) If n is small and Π 0.50: distribution is skewed to the. (B) If n is small and Π > 0.50: distribution is skewed to the. (C) If n is large and/ or Π = 0.50: distribution is.
67 Page 67 Eample: 30% of job applicants are from a minority group. If we take applicants at random, what is the probability of including from the minority group? Let X = minority person n=5; X=2; Π=0.3 n P( X ) = nc( Π) ( 1 Π) Binomial P( 2)! = (. ) (. ) = 0 2! 3!
68 Page 68 Eample: Customers in a bank. One in four customers require service for more than 4 minutes. people are in a queue. What is the probability that eactly 3 customers will take more than 4 minutes each? Π=0.25;n=6; =3 X= customer takes more than 4 minutes. P( X) = nc( Π) ( 1 Π) P() 3 n = (. )(. ). 6! = !!
69 Page 69 What is the probability that at least 3 customers need more than 4 minutes each? I.e. P(3)+ P(4) + P(5) + P(6): P( 4 6! 4 2 ) = (. 0 25)(. 0 75) = 42!! P( 5 6! 5 1 ) = (. 0 25)(. 0 75) = 51!! 6! 6 0 P( 6) = ( 0. 25) ( 0. 75) = 6!0! So the probability is: ( ) = 0. & P(< 3 customers take more than 4 minutes each)= = 0.
70 Page 70
71 Page 71 The Epected Value of a Binomial Random Variable: The number of successes in any given eperiment must equal the number of trials (n) times the probability of a success on each trial ( Π). For eample, the probability that a process produces a defective item is Π = 0. ; Then the mean number of defectives in trials is: (10)(0.25) =2.5.
72 Page 72 Let X be ~ B(n, Π). Then E(X)=n Π : Probability of success = Π number of successes epected from a single trial is: { 1( ) 0( 1 )} Π + Π = Π So, in "n" trials, epected n Π "successes." Binomial mean: E(X Binomial)=
73 Proof: E( X) = P( ) n = 0 n = nc ( Π) ( 1 Π) = 0 n = ( ) = = = = 1 n = 1 n! Π!( n )! n! Π ( 1)!( n )! n ( n 1)! nπ Π ( 1 Π) ( 1)!( n 1 ( 1))! = 1 ( nπ ) y= 0 m! y!( m [ y = 1; m = n 1] So, E( ) = ( nπ)( 1) = nπ m Π y)! n Y= B( m, Π ) ( 1 Π) ( 1 Π) y n n ( 1 Π) 1 ( n 1) ( 1) m y Page 73
74 Similarly, we can show that V(X)= Page 74 So the standard deviation of X: nπ( 1 Π) when X~ B(n, Π).
75 Page 75 Eample: If a manufacturing process is working properly, 10% of the items produced will be defective. We take a random sample of 15 items. Calculate the number of defectives: X = defective ~ B(15, 0.1) So, E() = n Π= (15)(0.1) = 1.5 One or two defectives epected. What is the deviation of the number of defectives? V(X) = n Π (1- Π)=(15)(0.1)(0.9)=1.35 So, the standard deviation (X)= 1.16
76 Page 76
Bivariate distributions
Bivariate distributions 3 th October 017 lecture based on Hogg Tanis Zimmerman: Probability and Statistical Inference (9th ed.) Bivariate Distributions of the Discrete Type The Correlation Coefficient
More informationCHAPTER 4 MATHEMATICAL EXPECTATION. 4.1 Mean of a Random Variable
CHAPTER 4 MATHEMATICAL EXPECTATION 4.1 Mean of a Random Variable The expected value, or mathematical expectation E(X) of a random variable X is the long-run average value of X that would emerge after a
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 informationChapter 2. Some Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables
Chapter 2 Some Basic Probability Concepts 2.1 Experiments, Outcomes and Random Variables A random variable is a variable whose value is unknown until it is observed. The value of a random variable results
More informationProbability Distribution. Stat Camp for the MBA Program. Debbon Air Seat Release
Stat Camp for the MBA Program Daniel Solow Lecture 3 Random Variables and Distributions 136 Probability Distribution Recall that a random variable is a quantity of interest whose value is uncertain and
More informationCh. 5 Joint Probability Distributions and Random Samples
Ch. 5 Joint Probability Distributions and Random Samples 5. 1 Jointly Distributed Random Variables In chapters 3 and 4, we learned about probability distributions for a single random variable. However,
More informationSDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 13: Expectation and Variance and joint distributions Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin
More informationJoint Distribution of Two or More Random Variables
Joint Distribution of Two or More Random Variables Sometimes more than one measurement in the form of random variable is taken on each member of the sample space. In cases like this there will be a few
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 informationA random variable is a quantity whose value is determined by the outcome of an experiment.
Random Variables A random variable is a quantity whose value is determined by the outcome of an experiment. Before the experiment is carried out, all we know is the range of possible values. Birthday example
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 informationIntroduction to Statistical Data Analysis Lecture 3: Probability Distributions
Introduction to Statistical Data Analysis Lecture 3: Probability Distributions James V. Lambers Department of Mathematics The University of Southern Mississippi James V. Lambers Statistical Data Analysis
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 informationEE 345 MIDTERM 2 Fall 2018 (Time: 1 hour 15 minutes) Total of 100 points
Problem (8 points) Name EE 345 MIDTERM Fall 8 (Time: hour 5 minutes) Total of points How many ways can you select three cards form a group of seven nonidentical cards? n 7 7! 7! 765 75 = = = = = = 35 k
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 informationStatistics for Economists Lectures 6 & 7. Asrat Temesgen Stockholm University
Statistics for Economists Lectures 6 & 7 Asrat Temesgen Stockholm University 1 Chapter 4- Bivariate Distributions 41 Distributions of two random variables Definition 41-1: Let X and Y be two random variables
More informationStatistics 427: Sample Final Exam
Statistics 427: Sample Final Exam Instructions: The following sample exam was given several quarters ago in Stat 427. The same topics were covered in the class that year. This sample exam is meant to be
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 informationBivariate Distributions
STAT/MATH 395 A - PROBABILITY II UW Winter Quarter 17 Néhémy Lim Bivariate Distributions 1 Distributions of Two Random Variables Definition 1.1. Let X and Y be two rrvs on probability space (Ω, A, P).
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 informationECON Fundamentals of Probability
ECON 351 - Fundamentals of Probability Maggie Jones 1 / 32 Random Variables A random variable is one that takes on numerical values, i.e. numerical summary of a random outcome e.g., prices, total GDP,
More informationDiscrete random variables and probability distributions
Discrete random variables and probability distributions random variable is a mapping from the sample space to real numbers. notation: X, Y, Z,... Example: Ask a student whether she/he works part time or
More informationLecture 10. Variance and standard deviation
18.440: Lecture 10 Variance and standard deviation Scott Sheffield MIT 1 Outline Defining variance Examples Properties Decomposition trick 2 Outline Defining variance Examples Properties Decomposition
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Problem Set 4: Solutions Fall 2007
UC Berkeley Department of Electrical Engineering and Computer Science EE 6: Probablity and Random Processes Problem Set 4: Solutions Fall 007 Issued: Thursday, September 0, 007 Due: Friday, September 8,
More informationProbability Distributions
Probability Distributions Series of events Previously we have been discussing the probabilities associated with a single event: Observing a 1 on a single roll of a die Observing a K with a single card
More informationRandom Variables. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 13 Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 8, 2013 2 / 13 Random Variable Definition A real-valued
More informationStochastic Models of Manufacturing Systems
Stochastic Models of Manufacturing Systems Ivo Adan Organization 2/47 7 lectures (lecture of May 12 is canceled) Studyguide available (with notes, slides, assignments, references), see http://www.win.tue.nl/
More informationII. The Binomial Distribution
88 CHAPTER 4 PROBABILITY DISTRIBUTIONS 進佳數學團隊 Dr. Herbert Lam 林康榮博士 HKDSE Mathematics M1 II. The Binomial Distribution 1. Bernoulli distribution A Bernoulli eperiment results in any one of two possible
More informationLecture Notes 2 Random Variables. Discrete Random Variables: Probability mass function (pmf)
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 informationRandom Variable. Discrete Random Variable. Continuous Random Variable. Discrete Random Variable. Discrete Probability Distribution
Random Variable Theoretical Probability Distribution Random Variable Discrete Probability Distributions A variable that assumes a numerical description for the outcome of a random eperiment (by chance).
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
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 Variable And Probability Distribution. Is defined as a real valued function defined on the sample space S. We denote it as X, Y, Z,
Random Variable And Probability Distribution Introduction Random Variable ( r.v. ) Is defined as a real valued function defined on the sample space S. We denote it as X, Y, Z, T, and denote the assumed
More informationIE 336 Seat # Name (clearly) < KEY > Open book and notes. No calculators. 60 minutes. Cover page and five pages of exam.
Open book and notes. No calculators. 60 minutes. Cover page and five pages of exam. This test covers through Chapter 2 of Solberg (August 2005). All problems are worth five points. To receive full credit,
More informationLecture 16 : Independence, Covariance and Correlation of Discrete Random Variables
Lecture 6 : Independence, Covariance and Correlation of Discrete Random Variables 0/ 3 Definition Two discrete random variables X and Y defined on the same sample space are said to be independent if for
More informationThe mean, variance and covariance. (Chs 3.4.1, 3.4.2)
4 The mean, variance and covariance (Chs 3.4.1, 3.4.2) Mean (Expected Value) of X Consider a university having 15,000 students and let X equal the number of courses for which a randomly selected student
More informationElements of Probability Theory
Short Guides to Microeconometrics Fall 2016 Kurt Schmidheiny Unversität Basel Elements of Probability Theory Contents 1 Random Variables and Distributions 2 1.1 Univariate Random Variables and Distributions......
More informationCMPSCI 240: Reasoning Under Uncertainty
CMPSCI 240: Reasoning Under Uncertainty Lecture 8 Prof. Hanna Wallach wallach@cs.umass.edu February 16, 2012 Reminders Check the course website: http://www.cs.umass.edu/ ~wallach/courses/s12/cmpsci240/
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 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 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 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 informationIAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 3-RANDOM VARIABLES
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 3-RANDOM VARIABLES VARIABLE Studying the behavior of random variables, and more importantly functions of random variables is essential for both the
More informationNotes for Math 324, Part 19
48 Notes for Math 324, Part 9 Chapter 9 Multivariate distributions, covariance Often, we need to consider several random variables at the same time. We have a sample space S and r.v. s X, Y,..., which
More information(y 1, y 2 ) = 12 y3 1e y 1 y 2 /2, y 1 > 0, y 2 > 0 0, otherwise.
54 We are given the marginal pdfs of Y and Y You should note that Y gamma(4, Y exponential( E(Y = 4, V (Y = 4, E(Y =, and V (Y = 4 (a With U = Y Y, we have E(U = E(Y Y = E(Y E(Y = 4 = (b Because Y and
More informationDiscrete Probability Refresher
ECE 1502 Information Theory Discrete Probability Refresher F. R. Kschischang Dept. of Electrical and Computer Engineering University of Toronto January 13, 1999 revised January 11, 2006 Probability theory
More informationJoint Probability Distributions, Correlations
Joint Probability Distributions, Correlations What we learned so far Events: Working with events as sets: union, intersection, etc. Some events are simple: Head vs Tails, Cancer vs Healthy Some are more
More informationProbability Dr. Manjula Gunarathna 1
Probability Dr. Manjula Gunarathna Probability Dr. Manjula Gunarathna 1 Introduction Probability theory was originated from gambling theory Probability Dr. Manjula Gunarathna 2 History of Probability Galileo
More informationNotes 12 Autumn 2005
MAS 08 Probability I Notes Autumn 005 Conditional random variables Remember that the conditional probability of event A given event B is P(A B) P(A B)/P(B). Suppose that X is a discrete random variable.
More informationChapter 2 Random Variables
Stochastic Processes Chapter 2 Random Variables Prof. Jernan Juang Dept. of Engineering Science National Cheng Kung University Prof. Chun-Hung Liu Dept. of Electrical and Computer Eng. National Chiao Tung
More informationTopic 2: Probability & Distributions. Road Map Probability & Distributions. ECO220Y5Y: Quantitative Methods in Economics. Dr.
Topic 2: Probability & Distributions ECO220Y5Y: Quantitative Methods in Economics Dr. Nick Zammit University of Toronto Department of Economics Room KN3272 n.zammit utoronto.ca November 21, 2017 Dr. Nick
More informationMATH 3510: PROBABILITY AND STATS June 15, 2011 MIDTERM EXAM
MATH 3510: PROBABILITY AND STATS June 15, 2011 MIDTERM EXAM YOUR NAME: KEY: Answers in Blue Show all your work. Answers out of the blue and without any supporting work may receive no credit even if they
More informationIntroduction to Probability and Statistics Slides 3 Chapter 3
Introduction to Probability and Statistics Slides 3 Chapter 3 Ammar M. Sarhan, asarhan@mathstat.dal.ca Department of Mathematics and Statistics, Dalhousie University Fall Semester 2008 Dr. Ammar M. Sarhan
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 informationRandom variables (discrete)
Random variables (discrete) Saad Mneimneh 1 Introducing random variables A random variable is a mapping from the sample space to the real line. We usually denote the random variable by X, and a value that
More informationMULTIVARIATE PROBABILITY DISTRIBUTIONS
MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined
More informationChapter 2. Probability
2-1 Chapter 2 Probability 2-2 Section 2.1: Basic Ideas Definition: An experiment is a process that results in an outcome that cannot be predicted in advance with certainty. Examples: rolling a die tossing
More informationJoint probability distributions: Discrete Variables. Two Discrete Random Variables. Example 1. Example 1
Joint probability distributions: Discrete Variables Two Discrete Random Variables Probability mass function (pmf) of a single discrete random variable X specifies how much probability mass is placed on
More informationCME 106: Review Probability theory
: Probability theory Sven Schmit April 3, 2015 1 Overview In the first half of the course, we covered topics from probability theory. The difference between statistics and probability theory is the following:
More informationProbability. Paul Schrimpf. January 23, UBC Economics 326. Probability. Paul Schrimpf. Definitions. Properties. Random variables.
Probability UBC Economics 326 January 23, 2018 1 2 3 Wooldridge (2013) appendix B Stock and Watson (2009) chapter 2 Linton (2017) chapters 1-5 Abbring (2001) sections 2.1-2.3 Diez, Barr, and Cetinkaya-Rundel
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 informationPreliminary Statistics. Lecture 3: Probability Models and Distributions
Preliminary Statistics Lecture 3: Probability Models and Distributions Rory Macqueen (rm43@soas.ac.uk), September 2015 Outline Revision of Lecture 2 Probability Density Functions Cumulative Distribution
More informationSUMMARY OF PROBABILITY CONCEPTS SO FAR (SUPPLEMENT FOR MA416)
SUMMARY OF PROBABILITY CONCEPTS SO FAR (SUPPLEMENT FOR MA416) D. ARAPURA This is a summary of the essential material covered so far. The final will be cumulative. I ve also included some review problems
More informationBayesian statistics, simulation and software
Module 1: Course intro and probability brush-up Department of Mathematical Sciences Aalborg University 1/22 Bayesian Statistics, Simulations and Software Course outline Course consists of 12 half-days
More informationECSE B Solutions to Assignment 8 Fall 2008
ECSE 34-35B Solutions to Assignment 8 Fall 28 Problem 8.1 A manufacturing system is governed by a Poisson counting process {N t ; t < } with rate parameter λ >. If a counting event occurs at an instant
More informationChapter 3. Discrete Random Variables and Their Probability Distributions
Chapter 3. Discrete Random Variables and Their Probability Distributions 2.11 Definition of random variable 3.1 Definition of a discrete random variable 3.2 Probability distribution of a discrete random
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 informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions
Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions 1999 Prentice-Hall, Inc. Chap. 4-1 Chapter Topics Basic Probability Concepts: Sample
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 informationLecture 1: Bayesian Framework Basics
Lecture 1: Bayesian Framework Basics Melih Kandemir melih.kandemir@iwr.uni-heidelberg.de April 21, 2014 What is this course about? Building Bayesian machine learning models Performing the inference of
More informationMgtOp 215 Chapter 5 Dr. Ahn
MgtOp 215 Chapter 5 Dr. Ahn Random variable: a variable that assumes its values corresponding to a various outcomes of a random experiment, therefore its value cannot be predicted with certainty. Discrete
More informationJointly Distributed Random Variables
Jointly Distributed Random Variables CE 311S What if there is more than one random variable we are interested in? How should you invest the extra money from your summer internship? To simplify matters,
More informationGEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs
STATISTICS 4 Summary Notes. Geometric and Exponential Distributions GEOMETRIC -discrete A discrete random variable R counts number of times needed before an event occurs P(X = x) = ( p) x p x =,, 3,...
More informationThe Binomial distribution. Probability theory 2. Example. The Binomial distribution
Probability theory Tron Anders Moger September th 7 The Binomial distribution Bernoulli distribution: One experiment X i with two possible outcomes, probability of success P. If the experiment is repeated
More informationDiscrete Probability Distribution
Shapes of binomial distributions Discrete Probability Distribution Week 11 For this activity you will use a web applet. Go to http://socr.stat.ucla.edu/htmls/socr_eperiments.html and choose Binomial coin
More informationmatrix-free Elements of Probability Theory 1 Random Variables and Distributions Contents Elements of Probability Theory 2
Short Guides to Microeconometrics Fall 2018 Kurt Schmidheiny Unversität Basel Elements of Probability Theory 2 1 Random Variables and Distributions Contents Elements of Probability Theory matrix-free 1
More informationLecture 2: Review of Basic Probability Theory
ECE 830 Fall 2010 Statistical Signal Processing instructor: R. Nowak, scribe: R. Nowak Lecture 2: Review of Basic Probability Theory Probabilistic models will be used throughout the course to represent
More informationACM 116: Lectures 3 4
1 ACM 116: Lectures 3 4 Joint distributions The multivariate normal distribution Conditional distributions Independent random variables Conditional distributions and Monte Carlo: Rejection sampling Variance
More informationSTA 584 Supplementary Examples (not to be graded) Fall, 2003
Page 1 of 8 Central Michigan University Department of Mathematics STA 584 Supplementary Examples (not to be graded) Fall, 003 1. (a) If A and B are independent events, P(A) =.40 and P(B) =.70, find (i)
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 informationRandom Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
More informationChapter 5 Joint Probability Distributions
Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter 5 Joint Probability Distributions 5 Joint Probability Distributions CHAPTER OUTLINE 5-1 Two
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 1
CS434a/541a: Pattern Recognition Prof. Olga Veksler Lecture 1 1 Outline of the lecture Syllabus Introduction to Pattern Recognition Review of Probability/Statistics 2 Syllabus Prerequisite Analysis of
More informationChapter 4. Probability-The Study of Randomness
Chapter 4. Probability-The Study of Randomness 4.1.Randomness Random: A phenomenon- individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.
More informationMath 218 Supplemental Instruction Spring 2008 Final Review Part A
Spring 2008 Final Review Part A SI leaders: Mario Panak, Jackie Hu, Christina Tasooji Chapters 3, 4, and 5 Topics Covered: General probability (probability laws, conditional, joint probabilities, independence)
More informationProbability. Table of contents
Probability Table of contents 1. Important definitions 2. Distributions 3. Discrete distributions 4. Continuous distributions 5. The Normal distribution 6. Multivariate random variables 7. Other continuous
More informationUNIT-2: MULTIPLE RANDOM VARIABLES & OPERATIONS
UNIT-2: MULTIPLE RANDOM VARIABLES & OPERATIONS In many practical situations, multiple random variables are required for analysis than a single random variable. The analysis of two random variables especially
More informationDependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline.
Practitioner Course: Portfolio Optimization September 10, 2008 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y ) (x,
More informationTutorial for Lecture Course on Modelling and System Identification (MSI) Albert-Ludwigs-Universität Freiburg Winter Term
Tutorial for Lecture Course on Modelling and System Identification (MSI) Albert-Ludwigs-Universität Freiburg Winter Term 2016-2017 Tutorial 3: Emergency Guide to Statistics Prof. Dr. Moritz Diehl, Robin
More informationReview of probability
Review of probability Computer Sciences 760 Spring 2014 http://pages.cs.wisc.edu/~dpage/cs760/ Goals for the lecture you should understand the following concepts definition of probability random variables
More informationSome Practice Questions for Test 2
ENGI 441 Probability and Statistics Faculty of Engineering and Applied Science Some Practice Questions for Test 1. The probability mass function for X = the number of major defects in a randomly selected
More informationEstadística I Exercises Chapter 4 Academic year 2015/16
Estadística I Exercises Chapter 4 Academic year 2015/16 1. An urn contains 15 balls numbered from 2 to 16. One ball is drawn at random and its number is reported. (a) Define the following events by listing
More informationECON 5350 Class Notes Review of Probability and Distribution Theory
ECON 535 Class Notes Review of Probability and Distribution Theory 1 Random Variables Definition. Let c represent an element of the sample space C of a random eperiment, c C. A random variable is a one-to-one
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 informationFinal Examination December 16, 2009 MATH Suppose that we ask n randomly selected people whether they share your birthday.
1. Suppose that we ask n randomly selected people whether they share your birthday. (a) Give an expression for the probability that no one shares your birthday (ignore leap years). (5 marks) Solution:
More informationSTAT 302 Introduction to Probability Learning Outcomes. Textbook: A First Course in Probability by Sheldon Ross, 8 th ed.
STAT 302 Introduction to Probability Learning Outcomes Textbook: A First Course in Probability by Sheldon Ross, 8 th ed. Chapter 1: Combinatorial Analysis Demonstrate the ability to solve combinatorial
More informationChapter 2: Discrete Distributions. 2.1 Random Variables of the Discrete Type
Chapter 2: Discrete Distributions 2.1 Random Variables of the Discrete Type 2.2 Mathematical Expectation 2.3 Special Mathematical Expectations 2.4 Binomial Distribution 2.5 Negative Binomial Distribution
More informationSTAT 430/510: Lecture 16
STAT 430/510: Lecture 16 James Piette June 24, 2010 Updates HW4 is up on my website. It is due next Mon. (June 28th). Starting today back at section 6.7 and will begin Ch. 7. Joint Distribution of Functions
More informationWeek 2: Review of probability and statistics
Week 2: Review of probability and statistics Marcelo Coca Perraillon University of Colorado Anschutz Medical Campus Health Services Research Methods I HSMP 7607 2017 c 2017 PERRAILLON ALL RIGHTS RESERVED
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 information