Chapter 5 Discrete Random Variables and Probability Distributions © Random Variables A random variable is a variable that takes on numerical values determined by the outcome of a random experiment. Discrete Random Variables A random variable is discrete if it can take on no more than a countable number of values. Discrete Random Variables (Examples) 1. The number of defective items in a sample of twenty items taken from a large shipment. 2. The number of customers arriving at a checkout counter in an hour. 3. The number of errors detected in a corporation’s accounts. 4. The number of claims on a medical insurance policy in a particular year. Continuous Random Variables A random variable is continuous if it can take any value in an interval. Continuous Random Variables (Examples) 1. The income in a year for a family. 2. The amount of oil imported into the U.S. in a particular month. 3. The change in the price of a share of IBM common stock in a month. 4. The time that elapses between the installation of a new computer and its failure. 5. The percentage of impurity in a batch of chemicals. Discrete Probability Distributions The probability distribution function (DPF), P(x), of a discrete random variable expresses the probability that X takes the value x, as a function of x. That is P( x) P( X x), for all values of x. Discrete Probability Distributions Graph the probability distribution function for the roll of a single six-sided die. P(x) 1/6 1 2 3 4 Figure 5.1 5 6 x Required Properties of Probability Distribution Functions of Discrete Random Variables Let X be a discrete random variable with probability distribution function, P(x). Then i. P(x) 0 for any value of x ii. The individual probabilities sum to 1; that is P( x) 1 x Where the notation indicates summation over all possible values x. Cumulative Probability Function The cumulative probability function, F(x0), of a random variable X expresses the probability that X does not exceed the value x0, as a function of x0. That is F ( x0 ) P( X x0 ) Where the function is evaluated at all values x0 Derived Relationship Between Probability and Cumulative Probability Function Let X be a random variable with probability function P(x) and cumulative probability function F(x0). Then it can be shown that F ( x0 ) P( X ) x x0 Where the notation implies that summation is over all possible values x that are less than or equal to x0. Derived Properties of Cumulative Probability Functions for Discrete Random Variables Let X be a discrete random variable with a cumulative probability function, F(x0). Then we can show that i. 0 F(x0) 1 for every number x0 ii. If x0 and x1 are two numbers with x0 < x1, then F(x0) F(x1) Expected Value The expected value, E(X), of a discrete random variable X is defined E ( X ) xP( x) x Where the notation indicates that summation extends over all possible values x. The expected value of a random variable is called its mean and is denoted x. Variance and Standard Deviation Let X be a discrete random variable. The expectation of the squared discrepancies about the mean, (X - )2, is called the variance, denoted 2x and is given by x2 E ( X x ) 2 ( x x ) 2 P( x) x The standard deviation, x , is the positive square root of the variance. Variance (Alternative Formula) The variance of a discrete random variable X can be expressed as E( X ) x 2 x 2 2 x P( x) x 2 x 2 Expected Value and Variance for Discrete Random Variable Using Microsoft Excel Sales 0 1 2 3 4 5 (Figure 5.4) P(x) Mean 0.15 0.00 0.30 0.30 0.20 0.40 0.20 0.60 0.10 0.40 0.05 0.25 1.95 Expected Value = 1.95 Variance 0.5704 0.2708 0.0005 0.2205 0.4203 0.4651 1.9475 Variance = 1.9475 Bernoulli Distribution A Bernoulli distribution arises from a random experiment which can give rise to just two possible outcomes. These outcomes are usually labeled as either “success” or “failure.” If denotes the probability of a success and the probability of a failure is (1 - ), the the Bernoulli probability function is P(0) (1 ) and P(1) Mean and Variance of a Bernoulli Random Variable The mean is: X E ( X ) xP( x) (0)(1 ) (1) X And the variance is: E[( X X ) ] ( x X ) P( x) 2 X 2 2 X (0 ) 2 (1 ) (1 ) 2 (1 ) Sequences of x Successes in n Trials The number of sequences with x successes in n independent trials is: n! C x!(n x)! n x Where n! = n x (x – 1) x (n – 2) x . . . x 1 and 0! = 1. These C xn sequencesare mutually exclusive, since no two of them can occur at the same time. Binomial Distribution Suppose that a random experiment can result in two possible mutually exclusive and collectively exhaustive outcomes, “success” and “failure,” and that is the probability of a success resulting in a single trial. If n independent trials are carried out, the distribution of the resulting number of successes “x” is called the binomial distribution. Its probability distribution function for the binomial random variable X = x is: P(x successes in n independent trials)= n! x ( n x) P( x) (1 ) x!(n x)! for x = 0, 1, 2 . . . , n Mean and Variance of a Binomial Probability Distribution Let X be the number of successes in n independent trials, each with probability of success . The x follows a binomial distribution with mean, X E( X ) n and variance, E[( X ) ] n (1 ) 2 X 2 Binomial Probabilities - An Example – (Example 5.7) An insurance broker, Shirley Ferguson, has five contracts, and she believes that for each contract, the probability of making a sale is 0.40. What is the probability that she makes at most one sale? P(at most one sale) = P(X 1) = P(X = 0) + P(X = 1) = 0.078 + 0.259 = 0.337 5! P(no sales) P(0) (0.4) 0 (0.6) 5 0.078 0!5! 5! P(1 sale) P(1) (0.4)1 (0.6) 4 0.259 1!4! Binomial Probabilities, n = 100, =0.40 (Figure 5.10) Sample size 100 Probability of success 0.4 Mean 40 Variance 24 Standard deviation 4.898979 Binomial Probabilities Table X 36 37 38 39 40 41 42 43 P(X) 0.059141 0.068199 0.075378 0.079888 0.081219 0.079238 0.074207 0.066729 P(<=X) 0.238611 0.30681 0.382188 0.462075 0.543294 0.622533 0.69674 0.763469 P(<X) 0.179469 0.238611 0.30681 0.382188 0.462075 0.543294 0.622533 0.69674 P(>X) 0.761389 0.69319 0.617812 0.537925 0.456706 0.377467 0.30326 0.236531 P(>=X) 0.820531 0.761389 0.69319 0.617812 0.537925 0.456706 0.377467 0.30326 Hypergeometric Distribution Suppose that a random sample of n objects is chosen from a group of N objects, S of which are successes. The distribution of the number of X successes in the sample is called the hypergeometric distribution. Its probability function is: C xS CnNxS P( x) N Cn S! ( N S )! x!( S x)! (n x)!( N S n x)! N! n!( N n)! Where x can take integer values ranging from the larger of 0 and [n-(N-S)] to the smaller of n and S. Poisson Probability Distribution 1) 2) 3) Assume that an interval is divided into a very large number of subintervals so that the probability of the occurrence of an event in any subinterval is very small. The assumptions of a Poisson probability distribution are: The probability of an occurrence of an event is constant for all subintervals. There can be no more than one occurrence in each subinterval. Occurrences are independent; that is, the number of occurrences in any non-overlapping intervals in independent of one another. Poisson Probability Distribution The random variable X is said to follow the Poisson probability distribution if it has the probability function: e x P( x) , for x 0, 1,2,... x! where 1. P(x) = the probability of x successes over a given period of time or space, given 2. = the expected number of successes per time or space unit; > 0 3. e = 2.71828 (the base for natural logarithms) Poisson Probability Distribution • The mean and variance of the Poisson probability distribution are: x E ( X ) and E[( X ) ] 2 x 2 Partial Poisson Probabilities for = 0.03 Obtained Using Microsoft Excel PHStat (Figure 5.14) Poisson Probabilities Table X P(X) 0 0.970446 1 0.029113 2 0.000437 3 0.000004 4 0.000000 P(<=X) 0.970446 0.999559 0.999996 1.000000 1.000000 P(<X) 0.000000 0.970446 0.999559 0.999996 1.000000 P(>X) 0.029554 0.000441 0.000004 0.000000 0.000000 P(>=X) 1.000000 0.029554 0.000441 0.000004 0.000000 Poisson Approximation to the Binomial Distribution Let X be the number of successes resulting from n independent trials, each with a probability of success, . The distribution of the number of successes X is binomial, with mean n. If the number of trials n is large and n is of only moderate size (preferably n 7), this distribution can be approximated by the Poisson distribution with = n. The probability function of the approximating distribution is then: P( x) e n (n ) , for x 0, 1,2,... x! x Joint Probability Functions Let X and Y be a pair of discrete random variables. Their joint probability function expresses the probability that X takes the specific value x and simultaneously Y takes the value y, as a function of x and y. The notation used is P(x, y) so, P ( x, y ) P ( X x Y y ) Joint Probability Functions Let X and Y be a pair of jointly distributed random variables. In this context the probability function of the random variable X is called its marginal probability function and is obtained by summing the joint probabilities over all possible values; that is, P ( x ) P ( x, y ) y Similarly, the marginal probability function of the random variable Y is P ( y ) P ( x, y ) x Properties of Joint Probability Functions • Let X and Y be discrete random variables with joint probability function P(x,y). Then 1. P(x,y) 0 for any pair of values x and y 2. The sum of the joint probabilities P(x, y) over all possible values must be 1. Conditional Probability Functions Let X and Y be a pair of jointly distributed discrete random variables. The conditional probability function of the random variable Y, given that the random variable X takes the value x, expresses the probability that Y takes the value y, as a function of y, when the value x is specified for X. This is denoted P(y|x), and so by the definition of conditional probability: P( x, y ) P( y | x) P( x) Similarly, the conditional probability function of X, given Y = y is: P( x, y ) P( x | y ) P( y ) Independence of Jointly Distributed Random Variables The jointly distributed random variables X and Y are said to be independent if and only if their joint probability function is the product of their marginal probability functions, that is, if and only if P( x, y ) P( x) P( y ) for all possible pairs of values x and y. And k random variables are independent if and only if P( x , x ,, x ) P( x ) P( x ) P( x ) 1 2 k 1 2 k Expected Value Function of Jointly Distributed Random Variables Let X and Y be a pair of discrete random variables with joint probability function P(x, y). The expectation of any function g(x, y) of these random variables is defined as: E[ g ( X , Y )] g ( x, y ) P ( x, y ) x y Stock Returns, Marginal Probability, Mean, Variance (Example 5.16) Y Return X Return 0% 0% 5% 10% 15% 0.0625 0.0625 0.0625 0.0625 5% 0.0625 0.0625 0.0625 0.0625 10% 0.0625 0.0625 0.0625 0.0625 15% 0.0625 0.0625 0.0625 0.0625 Table 5.6 Covariance Let X be a random variable with mean X , and let Y be a random variable with mean, Y . The expected value of (X X )(Y - Y ) is called the covariance between X and Y, denoted Cov(X, Y). For discrete random variables Cov( X , Y ) E[( X X )(Y Y )] ( x x )( y y ) P( x, y ) x y An equivalent expression is Cov( X , Y ) E ( XY ) x y xyP( x, y ) x y x y Correlation Let X and Y be jointly distributed random variables. The correlation between X and Y is: Corr ( X , Y ) Cov( X , Y ) XY Covariance and Statistical Independence If two random variables are statistically independent, the covariance between them is 0. However, the converse is not necessarily true. Portfolio Analysis The random variable X is the price for stock A and the random variable Y is the price for stock B. The market value, W, for the portfolio is given by the linear function, W aX bY Where, a, is the number of shares of stock A and, b, is the number of shares of stock B. Portfolio Analysis The mean value for W is, The variance for W is, W E[W ] E[aX bY ] a X bY a b 2abCov( X , Y ) 2 W 2 2 X 2 2 Y or using the correlation, a b 2abCorr ( X , Y ) X Y 2 W 2 2 X 2 2 Y