Discrete Random Variables and Probability Distributions 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) The number of defective items in a sample of twenty items taken from a large shipment. The number of customers arriving at a check-out counter in an hour. The number of errors detected in a corporation’s accounts. 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) The income in a year for a family. The amount of oil imported into the U.S. in a particular month. The change in the price of a share of IBM common stock in a month. The time that elapses between the installation of a new computer and its failure. 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

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 5 6 x Figure 5.1

Required Properties of Probability Distribution Functions of Discrete Random Variables Let X be a discrete random variable with probability distribution function, P(x). Then P(x)  0 for any value of x The individual probabilities sum to 1; that is 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 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 Where the notation implies that summation is over all possible values x that are less than or equal to x0.

0  F(x0)  1 for every number 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 0  F(x0)  1 for every number x0 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 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 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

Expected Value and Variance for Discrete Random Variable Using Microsoft Excel (Figure 5.4)

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

Mean and Variance of a Bernoulli Random Variable The mean is: And the variance is:

Sequences of x Successes in n Trials The number of sequences with x successes in n independent trials is: Where n! = n x (x – 1) x (n – 2) x . . . x 1 and 0! = 1.

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)= 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, and variance,

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

Binomial Probabilities, n = 100,  =0.40 (Figure 5.10)

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: 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 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: where P(x) = the probability of x successes over a given period of time or space, given   = the expected number of successes per time or space unit;  > 0 e = 2.71828 (the base for natural logarithms)

Poisson Probability Distribution The mean and variance of the Poisson probability distribution are:

Partial Poisson Probabilities for  = 0 Partial Poisson Probabilities for  = 0.03 Obtained Using Microsoft Excel PHStat (Figure 5.14)

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:

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,

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, Similarly, the marginal probability function of the random variable Y is

Properties of Joint Probability Functions Let X and Y be discrete random variables with joint probability function P(x,y). Then P(x,y)  0 for any pair of values x and y 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: Similarly, the conditional probability function of X, given Y = y is:

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 And k random variables are independent if and only if

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:

Stock Returns, Marginal Probability, Mean, Variance (Example 5.16) Y Return X Return 0% 5% 10% 15% 0.0625 Table 5.6

For discrete random variables An equivalent expression is 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 An equivalent expression is

Correlation Let X and Y be jointly distributed random variables. The correlation between X and Y is:

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, 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, or using the correlation,