1 Probability and Random Variables

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1 1 Probability and Random Variables The models that you have seen thus far are deterministic models. For any time t, there is a unique solution X(t). On the other hand, stochastic models will result in a distribution of possible values X(t) at a time t. To understand the properties of stochastic models, we need to use the language of probability and random variables. 1.1 The Basic Ideas of Probability Sample Spaces and Events Probability: Probability is used to make inferences about populations. Experiment: Some process whose outcome is not known with certainty. Sample Space: The collection of all possible outcomes of an experiment or process; denoted S. Event: Any collection of possible outcomes of an experiment; denoted A, B, etc. Relative Frequency Interpretation of Probability A random experiment is carried out a large number (n) of times and the number (n(a)) of times that event A occurs is recorded. Then the proportion of times that A occurs will tend to the probability of A: n(a) n P (A) Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 1

2 Illustration of Long-Run Relative Frequency Suppose a die is tossed repeatedly, and we count the number of times that the toss results in six spots. We then plot the proportion of times that the toss results in a six versus the number of tosses. n n(a) n(a) n Relative Frequency of Tosses of Die Resulting in a Six relative frequency n Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 2

3 Axioms: 1. P (A) > 0 for any event A 2. P (S) = 1 3. For any collection A 1, A 2,... of mutually exclusive events (A i A j = ), P (A 1 A 2 ) = P (A i ) i=1 Properties: 0 P (A) 1 P ( ) = 0 Probability an event does not occur: P (A ) = 1 P (A). P (A B) = P (A) + P (B) P (A B) If A and B are mutually exclusive, P (A B) = P (A) + P (B) Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 3

4 1.1.2 Conditional Probability For any two events A and B with P (B) > 0 the conditional probability of A given that B has occurred: P (A B) = The multiplication rule for P (A B) is: P (A B) P (B) Law of Total Probability P (A B) = P (A B)P (B) P (A B) = P (B A)P (A) Let A 1,..., A n be mutually exclusive and exhaustive events. Exhaustive means that A 1 A 2 A n = S. Assume also that P (A j ) > 0 for each j. Then for any event B, P (B) = n P (B A j )P (A j ) i=1 If P (B) > 0, this law implies Bayes Theorem: P (A k B) = P (B A k )P (A k ) n j=1 P (B A j)p (A j ) Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 4

5 Example: Diagnostic Testing. Define the two events: A= event that disease is present B= event that diagnostic test is positive We usually know the following: Prevalence of disease, say P (A) =.001 Sensitivity of test, say P (B A) = 0.95 Specificity of test, say P (B A ) = 0.90 We want to know, P (A B) or P (A B ) Solution: P (A B) = P (B A)P (A) P (B A)P (A)+P (B A )P (A ) = (0.95)(0.001) (0.95)(0.001)+(1 0.90)( ) = P (A B ) = P (B A )P (A ) P (B A )P (A )+P (B A)P (A) = (0.90)(0.999) (0.90)(0.999)+(1 0.95)(0.001) = Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 5

6 1.1.3 Independence Two events A and B are independent if P (A B) = P (A)P (B). They are dependent otherwise. When P (A) > 0 and P (B) > 0, this definition is equivalent to. P (A B) = P (A) and P (B A) = P (B) Extension of Independence to Several Events We say that A 1, A 2,..., A n are mutually independent if for every subset {i 1,..., i k } (k 2), we have P (A i1 A i2 A ik ) = P (A i1 )P (A i2 ) P (A ik ) We say that A 1, A 2,..., A n are pairwise independent if for every pair (i, j), i j. P (A i A j ) = P (A i )P (A j ) Pairwise independence does not imply mutual independence. Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 6

7 1.2 Random Variables Random variables help us to make a link between probability and numbers that we observe as data. A random variable (rv) is a numerical valued function defined on a sample space. A random variable X maps an outcome in a sample space to a numerical value. The probability that a rv X takes a value in the set A is given by P [X A] = P [X 1 (A)]. We use capital letters such as X or Y to denote random variables. Let s be an elementary outcome. A value, X(s), of X is denoted x. A random variable is discrete if it can take on a finite or countable number of values. A continuous random variable takes on an uncountable number of values. Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 7

8 1.3 Probability Distributions of a Discrete R.V. The probability distribution of a discrete r.v. is a list of the distinct values x of X together with the associated probabilities: p(x) = P (X = x) By P (X = x), we mean P (A x ) where A x = {s S : X(s) = x}. We can express p(x) as a function or in a table: x x 1 x 2 x 3... x k p(x) p(x 1 ) p(x 2 ) p(x 3 )... p(x k ) A function p(x) or p x is a probability mass function (pmf) of some random variable X if p(x) 0 all x all x i p(x i ) = 1 Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 8

9 An alternative way to represent a probability distribution is by using the cumulative distribution function (cdf): F (x) = P (X x) = p(y), < x < y:y x For a discrete random variable taking values on x 1 < x 2 < < x k, p(x j ) = F (x j ) F (x j 1 ), j = 2,..., k Parameters of Probability Distributions Suppose that for each value of α, p(x; α) is a probability distribution for a random variable X. Then α is said to be a parameter of the distribution. The collection of distributions {p(x; α) : α A} is called a parametric family of distributions. Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 9

10 1.3.2 Expected Values of Discrete RV Mean of a discrete RV The mean of a rv X is E[X] = µ = x D x p(x) where D is the set of possible values of X. The Expected Value of a Function of X The expected value of a function h(x) is: E[h(X)] = µ h(x) = x D h(x) p(x) If h(x) is a linear function in the form ax + b: E(aX + b) = ae(x) + b Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 10

11 Variance of a discrete R.V. The variance of a discrete R.V.: V (X) = σ 2 = σx 2 = E[(X u) 2 ] The standard deviation of X is σ = σ X = V (X) = σ 2 = SD(X) The Variance of a Linear Function If h(x) = ax + b then: V (ax + b) = a 2 V (X) = a 2 σ 2 Implications: V (ax) = a 2 V (X) SD(aX) = a SD(X) V (X + b) = V (X) SD(X + b) = SD(X) Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 11

12 1.4 Continuous Random Variables A continuous random variable can assume any value in an interval on the real line. The distribution of a continuous random variable is determined by the probability density function (pdf). The pdf of X is a function f(x) such that for any numbers a and b where a < b, P (a X b) = b a f(x)dx The graph of f(x) is often called a density curve. For f(x) to be a pdf it must satisfy: 1. f(x) 0 all x 2. f(x)dx = 1 (area under curve is 1). An alternative method of expressing the distribution of a continuous random variable is using the cumulative distribution function (cdf). The cdf of a continuous RV is defined as: Useful Properties: F (x) = P (X x) = P (a X b) = F (b) F (a) x f(y)dy If X is a continuous RV with pdf f(x) and cdf F (x), then at every x at which F (x) exists: F (x) = f(x) Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 12

13 1.4.1 Percentiles For 0 p 1 the (100p) th percentile of the distribution of a continuous RV X is a value x p such that p = F (x p ) Expected Values, Mean and Variance The expected value of a function h(x) for a continuous rv is: E[h(X)] = h(x) f(x)dx Some special cases: Mean: E[X] = µ = x f(x)dx Variance: E[(X µ) 2 ] = σ 2 = (x µ)2 f(x)dx Remember: E[(X µ) 2 ] = E[X 2 ] (E[X]) 2 = σ 2 Note: The properties of expectation and variance of linear functions also hold in the continuous case. Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 13

14 1.5 Joint Probability Distributions The joint cdf of two random variables X and Y is defined by We say that (X, Y ) is discrete if F (x, y) = P [X x, Y y]. P [(X, Y ) A] = (x,y) A p(x, y) where p(x, y) = P [X = x, Y = y] is the joint pmf of (X, Y ). We say that (X, Y ) are jointly continuous rvs if there exists a function called the joint pdf such that P [(X, Y ) A] = A f(x, y)dxdy The expectation of a function h(x, Y ) of (X, Y ) is E[h(X, Y )] = h(x, y)f(x, y)dxdy h(x, y)p(x, y) x y if X, Y continuous if X, Y discrete Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 14

15 The random variables X and Y are independent if P [X A, Y B] = P [X A] P [Y B] for any events A and B. This is equivalent to F (x, y) = F X (x)f Y (y), for all x, y for any rvs f(x, y) = f X (x)f Y (y), for all x, y for continuous rvs p(x, y) = p X (x)p Y (y), for all x, y for discrete rvs Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 15

16 1.6 Some Special Cases Poisson Distribution Consider these random variables: Number of telephone calls received per hour. Number of days school is closed due to snow. Number of trees in an area of forest. Number of bacteria in a culture. A random variable X, the number of events occurring during a given time interval or in a specified region, is called a Poisson random variable. The corresponding distribution: X Poisson(λ) where λ is the rate per unit time or rate per unit area. p(x; λ) = P (X = x) = e λ λ x x!, x = 0, 1, 2,..., λ > 0 The mean and variance of a Poisson random variable are E[X] = µ = λ V [X] = σ 2 = λ Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 16

17 1.6.2 The Poisson Process We will be examining various stochastic processes that correspond to some of the deterministic population models studied so far. A stochastic processes {X(t), t T } is an indexed collection of random variables. We will show how the Poisson distribution arises from a stochastic process for which we make a few reasonable assumptions. We first define a counting process. A stochastic process {X(t), t 0} is said to be a counting process if 1. X(t) 0 2. X(t) is integer valued 3. If s < t, then X(s) X(t). 4. For s < t, X(t) X(s) equals the number of events that have occurred in the interval (s, t]. Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 17

18 Let {X(t), t 0} be a counting process that satisfies 1. X(0) = 0 2. X(t) is independent of X(t + s) X(s) for any s, t > 0 (independent increments). 3. The distribution of X(t + s) X(s) depends only on t for any s, t > 0 (stationary increments). 4. P (X(t) = 1) = λh + o(h) 5. P (X(t) 2) = o(h) Then we can show that P x (t) = P (X(t) = x) = e λt (λ)t x, x = 0, 1, 2,... x! The process {X(t), t 0} is called a Poisson process. Outline of Proof Consider P 0 (t + h) = P 0 (t)p 0 (h) = P 0 (t)(1 λh) + o(h). Then P 0 (t + h) P 0 (t) h = λp 0 (t) + o(h) h. Let h 0 and obtain P 0(t) = λp 0 (t). This implies that P 0 (t) = e λt. Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 18

19 For x 1, P x (t + h) = P [N(t) = x, N(t + h) N(t) = 0] + P [N(t) = x 1, N(t + h) N(t) = 1] + P [N(t + h) = x, N(t + h) N(t) 2] = P x (t)p 0 (h) + P x 1 (t)p 1 (h) + o(h) = (1 λh)p x (t) + λhp x 1 (t) + o(h) Divide both sides by h and let h 0: P x(t) = λp x (t) + λp x 1 (t), x = 1, 2,.... The solution to this system of differential equations is P x (t) = e λt (λt) x x!, x = 1, 2,.... Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 19

20 1.6.3 Normal Distribution The normal or Gaussian distribution has the pdf: f(x; µ, σ) = 1 2π σ e (x µ)2 /2σ 2 < x < The mean and variance are E(X) = µ V (X) = σ 2 The shorthand for this family of distributions as: X N(µ, σ 2 ) Some Normal Distributions Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 20

21 Normal Distributions with Different Means density mu=-2 mu=0 mu= x Normal Distributions with Different Variances y sigma=1 sigma=0.5 sigma= x Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 21

22 1.7 Gamma Distribution The gamma distribution is a family of distributions that yields a wide variety of skewed distributions. It is often used to model the lifetime length of manufactured items. Central to the gamma distribution is the gamma function: Γ(α) = 0 x α 1 e x dx α > 0 Some properties of the gamma function: 1. α > 1, Γ(α) = (α 1)Γ(α 1). 2. If n is positive integer: Γ(n) = (n 1)! 3. Γ( 1 2 ) = π. Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 22

23 Using the properties of the gamma function, we obtain the pdf of the gamma (α, β) distribution: f(x; α, β) = 1 β α Γ(α) xα 1 e x/β x 0, α > 0, β > 0 The mean and variance of the gamma distribution are: E(X) = αβ V (X) = αβ 2 α is the shape parameter and β is the scale parameter. If β = 1 then we call this the standard gamma distribution. If α = 1, the distribution is the exponential distribution. Letting λ = 1/β, the pdf of the exponential distribution is given by f(x; λ) = λe λx x 0, λ > 0 Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 23

24 Standard Gamma Density Curves f(x) alpha=1 alpha=2 alpha=3 alpha=4 alpha=5 alpha= x Gamma Density Curves with Different Scales, Alpha=2 f(x) beta=1 beta=2 beta=4 beta=8 beta= x Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 24

25 1.7.1 Distribution of Elapsed Time in the Poisson Process Recall the Poisson distribution and how it is used to calculate probabilities for certain events in time or space: Let T 1 denote the time of the first event and T n, n = 2, 3,... be the time between the (n 1) st and n th events. Then (T 1, T 2,...) are independent and identically distributed (iid) exponential(λ) random variables. Let S n = T T n. Then S n has a gamma(1/λ, n) distribution. This can be noted by the fact that Hence, S n t N(t) n. P [S n t] = P [N(t) n] = j=n λt (λt)j e. j! We differentiate this to get the pdf of S n : f(t) = λn (n 1)! tn 1 e λt, t > 0. Chapter 1: Introduction to Probability and Random Variables Copyright c 2004 by Thomas E. Wehrly Slide 25

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