Discrete Probability Distribution: Definition, Types, and Examples

A comprehensive guide to discrete probability distributions, including definitions, types, examples, and their applications in statistics.

A discrete probability distribution is a statistical function that defines the probabilities of outcomes for discrete random variables. These variables take on a finite or countably infinite number of distinct values. The probability of each outcome must satisfy two conditions: each probability must be between 0 and 1, and the sum of all probabilities must equal 1.

Types of Discrete Probability Distributions

Binomial Distribution

The binomial distribution represents the number of successes in a fixed number of independent Bernoulli trials with the same probability of success. The probability mass function (PMF) is given by:

$$ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $$

where:

• \( n \) is the number of trials,
• \( k \) is the number of successes,
• \( p \) is the probability of success in each trial.

Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space. The PMF is given by:

$$ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$

where:

• \( \lambda \) is the average number of events in the interval,
• \( k \) is the actual number of events.

Geometric Distribution

The geometric distribution counts the number of Bernoulli trials needed to get one success. The PMF is given by:

$$ P(X = k) = (1-p)^{k-1} p $$

where:

• \( p \) is the probability of success,
• \( k \) is the trial on which the first success occurs.

Hypergeometric Distribution

The hypergeometric distribution describes successes in draws from a finite population without replacement. The PMF is given by:

$$ P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} $$

where:

• \( N \) is the population size,
• \( K \) is the number of successes in the population,
• \( n \) is the number of draws,
• \( k \) is the number of observed successes.

Examples of Discrete Probability Distributions

Example 1: Binomial Distribution

A quality control inspector checks 10 items for defects. If the probability of finding a defect in any item is 0.1, the distribution of defects follows a binomial distribution with \( n = 10 \) and \( p = 0.1 \).

Example 2: Poisson Distribution

A call center receives an average of 4 calls per minute. The number of calls received in any given minute follows a Poisson distribution with \( \lambda = 4 \).

Example 3: Geometric Distribution

The probability of a customer successfully completing a purchase on an e-commerce website is 0.2. The number of customers needed to get the first purchase follows a geometric distribution with \( p = 0.2 \).

Example 4: Hypergeometric Distribution

A deck of 52 cards contains 13 spades. If you draw 5 cards at random, the number of spades in your hand follows a hypergeometric distribution with \( N = 52 \), \( K = 13 \), and \( n = 5 \).

Applications and Special Considerations

Discrete probability distributions are crucial in various fields such as engineering, economics, finance, and social sciences. They help in modeling real-world scenarios where outcomes are distinct and countable.

Comparing Distributions

• Binomial vs. Poisson: The binomial distribution is used when the number of trials is fixed, while the Poisson distribution is concerned with the number of events in a continuous interval.
• Geometric vs. Binomial: The geometric distribution focuses on the first success, whereas the binomial distribution focuses on the number of successes in a fixed number of trials.

Related Terms

• Probability Mass Function (PMF): A function that gives the probability that a discrete random variable is exactly equal to a particular value.
• Random Variable: A variable whose value is subject to variations due to chance.

FAQs

What distinguishes discrete from continuous distributions?

Discrete distributions involve distinct, countable outcomes, while continuous distributions involve outcomes that can take any value within a range.

Can a probability be greater than 1?

No, the probability of any outcome in a discrete distribution must be between 0 and 1.

How do you check if a distribution is discrete or continuous?

Determine if the variable takes distinct values (discrete) or any value within a range (continuous).

References

1. Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
2. Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury Press.
3. Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2007). Mathematical Statistics with Applications. Thomson Brooks/Cole.

Summary

Discrete probability distributions play a vital role across various domains by modeling scenarios with specific, countable outcomes. Understanding the different types and their applications helps in solving practical problems effectively.