Discrete Probability Distributions

Introduction to Discrete Probability Distributions

Discrete probability distributions describe the probabilities of outcomes for discrete random variables, which take on whole number values. These distributions are foundational in statistics for modeling countable outcomes, such as the number of successes in a series of trials.

• Discrete Data: Values are integers, usually counted (e.g., number of complaints per day).

• Continuous Data: Values can take any value within a range, often measured (e.g., temperature, time).

Probability Distributions

Probability distributions can be classified as discrete or continuous. Discrete probability distributions are covered in this chapter, while continuous distributions are addressed separately.

• Discrete Probability Distribution: Lists all possible outcomes for a discrete random variable and their probabilities.

• Continuous Probability Distribution: Describes probabilities for continuous random variables.

Rules for Discrete Probability Distributions

• Each outcome must be mutually exclusive.

• Probabilities must satisfy: for all .

• The sum of all probabilities must be 1: .

Example: Probability Distribution for Coin Tosses

Consider tossing two coins and counting the number of heads. The probability distribution is:

Mean, Variance, and Standard Deviation of a Discrete Probability Distribution

• Mean (Expected Value):

• Variance:

• Standard Deviation:

For the coin toss example, the mean number of heads is 1.00, variance is 0.50, and standard deviation is approximately 0.707.

Comparing Distributions

The mean and standard deviation are useful for comparing different probability distributions. For example, comparing the number of rings before a call is answered at two call centers:

The Boston call center has a lower mean (faster answering) and is more consistent (lower standard deviation).

Expected Monetary Value (EMV)

The EMV is the mean of a discrete probability distribution when outcomes are measured in monetary terms. It represents the long-term average value if the experiment is repeated many times.

Binomial Distributions

Characteristics of a Binomial Experiment

• Fixed number of trials ()

• Each trial has two possible outcomes: success or failure

• Probability of success () and failure () are constant

• Trials are independent

Examples: Survey responses (yes/no), defective/acceptable items, job offer acceptance/rejection.

Binomial Probability Formula

The probability of exactly successes in trials is:

• = number of trials

• = number of successes

• = probability of success

• = probability of failure ()

Example: Binomial Probability Calculation

Suppose 40% of voters support Proposition A. In a sample of 10 voters, what is the probability exactly 5 support it?

Mean and Standard Deviation of a Binomial Distribution

• Mean:

• Standard Deviation:

For , mean is 4, standard deviation is approximately 1.549.

Using Binomial Probability Tables and Software

Binomial probabilities can be found using tables or software such as Excel or PHStat.

• Excel: =BINOM.DIST(x, n, p, cumulative)

• PHStat: Menu navigation to Binomial Probability Distribution

Poisson Distributions

Characteristics of a Poisson Process

• Counts the number of occurrences of an event over a fixed interval (time, area, etc.)

• The mean number of occurrences () is constant for each interval

• Occurrences in different intervals are independent

• Intervals do not overlap

Examples: Number of customers per hour, flaws per meter of cloth, accidents per month.

Poisson Probability Formula

The probability of exactly occurrences in an interval is:

• = mean number of occurrences

The variance of a Poisson distribution is equal to its mean:

Example: Poisson Probability Calculation

If a bank receives an average of 4 bad checks per week, what is the probability it receives exactly 3 next week?

Using Poisson Probability Tables and Software

Poisson probabilities can be found using tables or software such as Excel or PHStat.

• Excel: =POISSON.DIST(x, \lambda, cumulative)

• PHStat: Menu navigation to Poisson Probability Distribution

Poisson Approximation to the Binomial

The Poisson distribution can approximate the binomial distribution when and . The approximation is:

The Hypergeometric Distribution

Characteristics and Formula

The hypergeometric distribution is used when sampling is done without replacement from a finite population, making the trials dependent.

The probability of successes in a sample of size from a population of size with successes is:

• = population size

• = number of successes in population

• = sample size

• = number of successes in sample

Example: Hypergeometric Probability Calculation

Suppose 5 of 50 accounts are delinquent. If 10 accounts are sampled without replacement, what is the probability at least one is delinquent?

• Find

• , so

Mean and Standard Deviation of the Hypergeometric Distribution

• Mean:

• Standard Deviation:

Using Hypergeometric Probability Tables and Software

Hypergeometric probabilities can be calculated using Excel or PHStat.

• Excel: =HYPGEOM.DIST(x, n, R, N, cumulative)

• PHStat: Menu navigation to Hypergeometric Probability Distribution