Slide 1

Slide 2 1 EE571 PART 2 Probability and Random Variables Huseyin Bilgekul Eeng571 Probability and astochastic Processes Department of Electrical and Electronic Engineering Eastern Mediterranean University Slide 3 2 EE571 Chapter 4 Distribution Functions and Discrete Random Variables 4.1 Random Variables 4.2 Distribution Functions 4.3 Discrete Random Variables 4.4 Expectations of Discrete Random Variables 4.5 Variances and Moments of Discrete Random Variables 4.6 Standardized Random Variables Slide 4 3 EE571 4.1 Random Variables Definition Let S be the sample space of an experiment. A real- valued function X S R is called a random variable of the experiment if, for each interval I R, { s X(s) I } is an event. Example If in rolling two fair dice, X is the sum, then X can only assume the values 2, 3, 4, , 12 with the following probabilities P(X=2) = P({(1,1)}) =, P(X=3) = P({(1,2), (2,1)}) = P(X=4) = P({(1,3), (2,2), (3,1)}) = and, similarly Sum, s56789101112 P(X = s) Slide 5 4 EE571 Another Definition Definition A random variable X is a process of assigning a number X(s) to every outcome s of an experiment. The resulting function must satisfy the following two conditions but is otherwise arbitrary 1. The set {X x} is an event for every x. 2. The probabilities of the events {X = } and {X = - } equal 0: P{X = } = 0, P{X = - } = 0. P.S. X(s) is a real-valued function X S R Slide 6 5 EE571 Example 4.1 Suppose that 3 cards are drawn from an ordinary deck of 52 cards, one by one, at random and with replacement. Let X be the number of spades drawn; then X is a random variable. If an outcome of spades is denoted by s, and other outcomes are represented by t, then X is a real- valued function defined on the sample space S={(s,s,s), (t,s,s), (s,t,s), (s,s,t), (t,t,s), (t,s,t), (s,t,t), (t,t,t)} X(s,s,s) = 3, X(t,s,s) = X(s,s,t) = X(s,t,s) = 2, X(t,t,s) = X(s,t,t) = X(t,s,t) = 1, X(t,t,t) = 0, Slide 7 6 EE571 Example 4.1 (Contd) What are the probabilities of X = 0, 1, 2, 3 ? Sol Slide 8 7 EE571 Example 4.2 A bus stops at a station every day at some random time between 11:00 AM and 11:30 AM. If X is the actual arrival time of the bus, X is a random variable. It is defined on the sample space Then Slide 9 8 EE571 Example 4.3 In the United States, the number of twin births is approximately 1 in 90. Let X be the number of births in a certain hospital until the first twins are born. X is a random variable. Denote twin births by T and single births by N. Then X is a real-valued function defined on the sample space The set of all possible values of X is {1, 2, 3, } and Slide 10 9 EE571 Example 4.4 In a certain country, the draft-status priorities of eligible men are determined according to their birthdays. Numbers 1 to 366 are assigned to men with birthdays on Jan 1 to Dec 31. Then numbers are selected at random, one by one and without replacement, from 1 to 366 until all of them are chosen. Those with birthdays corresponding to the 1st number drawn would have the highest draft priority, those with birthdays corresponding to the 2nd number drawn have the 2nd-highest priority, and so on. Slide 11 10 EE571 Example 4.4 (Contd) Let X be the largest of the first 10 numbers selected. Then X is a random variable that assume the values 10, 11, 12, , 366. The event X = i occurs if the largest number among the first 10 is i, that is, if one of the first 10 numbers is i and the other 9 are from 1 through i 1. Thus, Slide 12 11 EE571 Example 4.5 The diameter of the metal disk manufactured by a factory is a random number between 4 and 4.5. What is the probability that the area of such a flat disk chosen at random is at least 4.41 ? Sol Ans: 3/5 Slide 13 12 EE571 Example 4.6 A random number is selected from the interval (0, /2). What is the probability that its sine is greater than its cosine? Sol Ans: 1/2 Slide 14 13 EE571 4.2 Distribution Functions Definition If X is a random variable, then the function F defined on ( , ) by F(t)=P(X t) is called the distribution function or cumulative distribution function (CDF) of X. Properties 1. F is nondecreasing. 2. lim t F(t) = 1. 3. lim t F(t) = 0. 4. F is right continuous. F(t+)=F(t) Slide 15 14 EE571 Properties of CDF 1.P(X > a) = 1 F(a) 2.P(a < X b) = F(b) F(a) 3.P(X < a) = lim n F(a 1/n) F(a ) 4.P(X a) = 1 F(a ) 5.P(X = a) = F(a) F(a ) Slide 16 15 EE571 Example 4.7 The distribution function of a random variable X is given by Compute the following quanties (a) P(X < 2) (b) P(X = 2) (c) P(1 X < 3) (d) P(X > 3/2) (e) P(X = 5/2) (f) P(2 0, let X n be the amount of"> 31 EE571 Example 4.19 Let X 0 be the amount of rain that will fall in the United States on the next Christmas day. For n > 0, let X n be the amount of rain that will fall in the United States on Christmas n years later. Let N be the smallest number of years that elapse before we get a Christmas rainfall greater than X 0. Suppose that P(X i = X j ) = 0 if i j, the events concerning the amount of rain on Christmas days of different years are all independent, and the X n s are identically distributed. Find the expected value of N. Slide 33 32 EE571 Example 4.19 (Contd) Sol Slide 34 33 EE571 Example 4.20 The tanks of a country s army are numbered 1 to N. In a war this country loses n random tanks to the enemy, who discovers that the captured tanks are numbered. If X 1, X 2, , X n are the numbers of the captured tanks, what is E(max X i ) ? How can the enemy use E(max X i ) to find an estimate of N, the total number of this country s tanks? Sol Slide 35 34 EE571 Example 4.22 (Polyas Urn Model) An urn contains w white and b blue chips. A chip is drawn at random and then is returned to the urn along with c > 0 chips of the same color. Prove that if n = 2, 3, 4, , such experiments are made, then at each draw the probability of a white chip is still w/(w+b). and the probability of a blue chip is b/(w+b). Pf Slide 36 35 EE571 Example 4.21 An urn contains w white and b blue chips. A chip is drawn at random and then is returned to the urn along with c > 0 chips of the same color. This experiment is then repeated successively. Let X n be the number of white chips drawn during the first n draws. Show that E(X n ) = nw/(w+b). Pf Binomial Distri. Slide 37 36 EE571 Theorem 4.1 If X is a constant random variable, that is, if P(X = c) = 1 for a constant c, then E(X) = c. Pf Slide 38 37 EE571 Theorem 4.2 Let X be a discrete random variable with set of possible values A and probability mass function p(x), and let g be a real-valued function. Then g(X) is a random variable with Pf Slide 39 38 EE571 Corollary Let X be a discrete random variable; g 1, g 2, , g n be real-valued functions, and let 1, 2, , n be real numbers. Then Pf Slide 40 39 EE571 Example 4.23 The probability mass function of a discrete random variable X is given by What is the expected value of X(6 X) ? Sol Ans 7 Slide 41 40 EE571 Example 4.24 A box contains 10 disks of radii 1, 2, , and 10, respectively. What is the expected value of the area of a disk selected at random from this box? Sol Ans:38.5 Slide 42 41 EE571 Example 4.25 (Investment) Let X be the amount paid to purchase an asset, and let Y be the amount received from the sale of the same asset. Putting fixed-income securities aside, the ratio Y/X is a random variable called the total return and is denoted by R. Obviously, Y = RX. The ratio r = (Y X)/X is a random variable called the rate of return. Clearly, r = (Y / X) 1 = R 1, or R = 1 + r. Let X be the total investment. Suppose that the portfolio of the investor consists of a total of n financial assets. Let w i be the fraction of investment in the i-th financial asset. Then X i = w i X is the amount invested in the i-th financial asset, and w i is called the weight of asset i. Slide 43 42 EE571 4.5 Variances and Moments of Discrete R.V. Definition Let X be a discrete random variable with a set of possible values A and probability mass function p(x), and E(X) = . Then Var(X) and X, called the variance and the standard deviation of X, respectively, are defined by Slide 44 43 EE571 Example 4.26 Two games Bolita and Keno. To play Bolita, you buy a ticket for $1, draws a ball at random from a box of 100 balls numbered 1 to 100. If the ball draw matches the number on your ticket, you win $75; otherwise, you lose. To play Keno, you bet $1 on a single number that has a 25% chance to win. If you win, they will return you dollar plus two dollars more; other, they keep the dollar. Let B and K be the amounts that you gain in one play of Bolita and Keno, respectively. Find the means and variances for B and K. Slide 45 44 EE571 Example 4.26 (Contd) Sol Ans: E(B) = 0.25, E(K) = 0.25 Var(B) = 55.69, Var(K) = 1.6875 Slide 46 45 EE571 Theorem 4.3 Pf Slide 47 46 EE571 Example 4.27 What is the variance of the random variable X, the outcome of rolling a fair die? Sol Ans: 35/12 Slide 48 47 EE571 Theorem 4.4 Let X be a discrete random variable with the set of possible values A and mean . Then Var(X) = 0 if and only if X is a constant with probability 1. Pf Prove it by contradiction. contradiction There does not exist any k p(k) >0. Slide 49 48 EE571 Theorem 4.5 Let X be a discrete random variable; then for constants a and b we have that Pf Slide 50 49 EE571 Example 4.28 Suppose that, for a discrete random variable X, E(X) = 2 and E[X(X 4)] = 5. Find the variance and the standard deviation of 4X +12. Sol Ans: Var( 4X+12) = 144 Slide 51 50 EE571 Concentration Definition Let X and Y be two random variables and be a given point. If for all t > 0, Then we say that X is more concentrated about than is Y. Theorem 4.6 Suppose that X and Y are two random variables with E(X) = E(Y) = . If X is more concentrated about than is Y, then Var(X) Var(Y). Slide 52 51 EE571 Moments Definition E[g(X)] Definition E(X n ) The nth moment of X E(|X| r ) The rth absolutemoment of X E(X c) The first moment of X about c E[(X c) n ] The nth moment of X about c E[(X ) n ] The nth central moment of X about E[X(X 1) (X k)] The factorial kth moment of X Remark 4.2 The existence of higher moments implies the existence of lower moments. Slide 53 52 EE571 4.6 Standardized Random Variables Definition The random variable is called the standardized X. Note: If X 1 = X+ , then X 1 * = X *.