Discrete Probability Distribution: Definition & Examples

Probability Distributions > Discrete Probability Distribution

Contents:

1. What is a Discrete Probability Distribution?
2. Discrete Probability Distribution Examples
3. Common discrete distributions
4. Properties

What is a Discrete Probability Distribution?

A discrete probability distribution is a set of probabilities associated with the outcomes of a random variable.

A discrete probability distribution is made up of discrete variables. Discrete variables are those which have a countable number of possible values. Examples include the outcome of a coin toss (heads or tails), the result of a dice roll (1 through 6), or the number of people who visit a website in one day (any integer from 0 to infinity).

Specifically, if a random variable is discrete, then it will have a discrete probability distribution.

Discrete Probability Distribution Examples

For example, let’s say you had the choice of playing two games of chance at a fair.

1. Game 1: Roll a die. If you roll a six, you win a prize.
2. Game 2: Guess the weight of the man. If you guess within 10 pounds, you win a prize.

One of these games is a discrete probability distribution and one is a continuous probability distribution. Which is which?

For game 1, you could roll a 1,2,3,4,5, or 6. All of the die rolls have an equal chance of being rolled (one out of six, or 1/6). This gives you a discrete probability distribution of:

Roll	1	2	3	4	5	6
Odds	1/6	1/6	1/6	1/6	1/6	1/6

For the guess the weight game, you could guess that the mean weighs 150 lbs. Or 210 lbs. Or 185.5 lbs. Or any fraction of a pound (172.566 pounds). Even if you stick to, say, between 150 and 200 lbs., the possibilities are endless:

• 160.1 lbs.
• 160.11 lbs.
• 160.111 lbs.
• 160.1111 lbs.
• 160.111111 lbs.

In reality, you probably wouldn’t guess 160.111111 lbs…that seems a little ridiculous. But it doesn’t change the fact that you could (if you wanted to), so that’s why it’s a continuous probability distribution.

Examples of Real-World Uses

1. Quality Control: Number of defective items in a batch (often modeled by Hypergeometric or Binomial distributions).
2. Queuing Theory: Number of arrivals in a time interval (often modeled by Poisson distribution).
3. Risk Analysis: Number of claims an insurance company might receive in a day (often Poisson).

Common discrete distributions

The following are examples of discrete probability distributions commonly used in statistics:

• Binomial distribution: Evaluates the probability for an outcome to either succeed or fail.
• Bernoulli distribution: A special case of the Binomial distribution with only one trial (e.g., success/failure).
• Discrete uniform distribution: All outcomes in a finite set have the same probability (like rolling a fair die).
• Geometric Distribution: Represents the number of failures before you get a success in a series of Bernoulli trials.
• Hypergeometric distribution: The hypergeometric distribution describes the probability of obtaining k successes in n draws, without replacement, from a finite population of size N containing exactly K successes.
• Multinomial Distribution: Used to find probabilities in experiments where there are more than two outcomes.
• Negative binomial distribution: A series of Bernoulli trials with constant success probability that continues until a specified number of successes is reached.
• Poisson distribution: gives us the probability of a given number of events happening in a fixed interval of time.

Properties

1. Probability Mass Function (PMF)

A PMF p(x) or a discrete random variable gives the probability that takes on a specific value x.

We can state that as p(x) = P(X = x).

For a PMF to be valid, these two conditions must be met:

1. p(x) ≥ 0 for all x in the domain.
2. The sum of the probabilities over all possible values must equal 1.

2. Cumulative Distribution Function (CDF)

Even though discrete variables often use a PMF to describe their probabilities, you can also define a cumulative distribution function (CDF), () = ( ≤ ) F(x) = P(X ≤ x).

For discrete variables, the CDF changes its value only at the discrete points where the random variable has positive probability.

3. Support of the Random Variable

• The support of a discrete random variable is the set of all values (finite or countably infinite) where the PMF is nonzero.
• A discrete variable can take either a finite or a countably infinite set of values (e.g., 0, 1, 2, …).

4. Expectation and Variance

For a discrete random variable X with PMF (), the expected value (mean) is

E[X] = Σ_x x p(x)

The variance is

Var(X) = E[X²] – (E[X])².

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