Chapter 5

Discrete Probability Distributions

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

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Discrete Probability Distributions

Graph the probability distribution function for the roll of a single six-sided die.

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P(x)

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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

1. P(x) ≥ 0 for any value of x
2. The individual probabilities sum to 1; that is

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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

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Where the function is evaluated at all values x₀

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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

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Where the notation implies that summation is over all possible values x that are less than or equal to x₀.

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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

1. 0 ≥ F(x₀) ≥ 1 for every number x₀
2. If x₀ and x₁ are two numbers with x₀ < x₁, then F(x₀) ≤ F(x₁)

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Expected Value

The expected value, E(X), of a discrete random variable X is defined

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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 μ_x.

Variance and Standard Deviation

Let X be a discrete random variable. The expectation of the squared discrepancies about the mean, (X - μ)², is called the variance, denoted σ²_x and is given by

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The standard deviation, σ_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)

Expected Value = 1.95

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 π denotes the probability of a success and the probability of a failure is (1 - π ), the the Bernoulli probability function is

Mean and Variance of a Bernoulli Random Variable

The mean is:

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And the variance is:

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Sequences of x Successes in n Trials

The number of sequences with x successes in n independent trials is:

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Where n! = n x (x – 1) x (n – 2) x . . . x 1 and 0! = 1.

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Binomial Distribution

Suppose that a random experiment can result in two possible mutually exclusive and collectively exhaustive outcomes, “success” and “failure,” and that π 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)=

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for x = 0, 1, 2 . . . , n

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Mean and Variance of a Binomial Probability Distribution

Let X be the number of successes in n independent trials, each with probability of success π. The x follows a binomial distribution with mean,

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and variance,

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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, π =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:

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Where x can take integer values ranging from the larger of 0 and [n-(N-S)] to the smaller of n and S.

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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:

1. The probability of an occurrence of an event is constant for all subintervals.
2. There can be no more than one occurrence in each subinterval.
3. 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:

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where

1. P(x) = the probability of x successes over a given period of time or space, given λ
2. λ = the expected number of successes per time or space unit; λ > 0
3. e = 2.71828 (the base for natural logarithms)

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Poisson Probability Distribution

• The mean and variance of the Poisson probability distribution are:

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Partial Poisson Probabilities for λ = 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, π. The distribution of the number of successes X is binomial, with mean nπ. If the number of trials n is large and nπ is of only moderate size (preferably nπ ≤ 7), this distribution can be approximated by the Poisson distribution with λ = nπ. The probability function of the approximating distribution is then:

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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,

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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.

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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:

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Similarly, the conditional probability function of X, given Y = y is:

Stock Returns, Marginal Probability, Mean, Variance�(Example 5.16)

Table 5.6

Covariance

Let X be a random variable with mean μ_X , and let Y be a random variable with mean, μ_Y . The expected value of (X - μ_X )(Y - μ_Y ) is called the covariance between X and Y, denoted Cov(X, Y).

For discrete random variables

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An equivalent expression is

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Correlation

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

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Covariance and Statistical Independence

If two random variables are statistically independent, the covariance between them is 0. However, the converse is not necessarily true.

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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,

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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,

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The variance for W is,

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or using the correlation,

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