What is a discrete probability distribution

Discrete versus continuous probability distributions

Continuous probability distributions apply to random variables that can assume any value within a range, such as heights, weights, or time measurements. This leads to an infinite number of possibilities within any given interval.

Discrete distributions use probability mass functions to compute the chance of specific outcomes directly. Continuous distributions rely on probability density functions, where the probability of any exact value equals zero, and probabilities are calculated over intervals instead.

Discrete variable: Number of heads in coin flips (0, 1, 2, 3, and so on).

Continuous variable: Exact weight of a person within a range of 150 to 250 pounds.

What conditions must a discrete probability distribution satisfy?

A discrete probability distribution is valid when two conditions are met. Each probability assigned to a possible outcome must be between 0 and 1, inclusive. The sum of all probabilities across every possible outcome must equal exactly 1.

These conditions are expressed mathematically as follows:

$0\le P(X=x)\le 1\text{for all outcomes }x$

$\sum _{x}P(X=x)=1$

These properties ensure the distribution represents a complete and feasible model of probabilities for a discrete random variable. Violation of either condition invalidates the distribution, as probabilities cannot exceed certainty or leave outcomes unaccounted for.

Verification through example

Consider a standard six-sided die roll. Each face (1 through 6) has a probability of 1/6. Each probability satisfies the condition 0 ≤ 1/6 ≤ 1, and the sum equals 1 (since 6 × 1/6 = 1). This makes the distribution valid. Changing one probability to 1/5 would cause the sum to exceed 1, rendering the distribution invalid.

What is a probability mass function?

A probability mass function (PMF) assigns the probability that a discrete random variable equals a specific value. The PMF is denoted as:

$p(x)=P(X=x)$

  This function specifies the probability for each possible outcome x in the distribution.

Relationship to discrete distributions

The PMF fully defines a discrete probability distribution by specifying probabilities for all countable outcomes. The probability mass function gives the probability of a discrete random variable X being exactly equal to some value x. The hypergeometric distribution is another commonly used type.

Bernoulli distribution

The Bernoulli distribution models a single trial with two outcomes: success (probability p) or failure (probability 1 − p). It applies to simple yes/no events, such as a coin flip landing heads.

$P(X=x)=p^x(1-p{)}^{1-x}\text{for }x\in \{0,1\}$

Binomial distribution

The binomial distribution counts the number of successes in a fixed number of independent Bernoulli trials, each with success probability p. The binomial distribution evaluates the probability for an outcome to either succeed or fail. Common applications include counting defects in sampled items or correct answers on a quiz.

$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$

Where n is the number of trials and k is the number of successes.

Poisson distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space, given an average rate λ. It fits rare, independent events such as customer arrivals per hour, emails received daily, or number of earthquakes in a time period.

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

Geometric distribution

The geometric distribution describes the number of trials until the first success in independent Bernoulli trials. Applications include rolls of a die until the first six appears or customer inquiries until a sale occurs.

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

Negative binomial distribution

The negative binomial distribution extends the geometric distribution to count the number of trials until the r-th success. It applies to repeated trials needing multiple successes, such as sales calls until three deals close.

$P(X=k)=(\genfrac{}{}{0ex}{}{k-1}{r-1})p^r(1-p{)}^{k-r}$

Hypergeometric distribution

The hypergeometric distribution counts successes in draws without replacement from a finite population. It is often used in quality control for modeling the number of defective items in a batch. Other applications include counting red cards drawn from a deck.

$P(X=k)=\frac{(\genfrac{}{}{0ex}{}{K}{k})(\genfrac{}{}{0ex}{}{N-K}{n-k})}{(\genfrac{}{}{0ex}{}{N}{n})}$

Where N is the population size, K is the number of success states in the population, n is the number of draws, and k is the number of observed successes.

Discrete uniform distribution

The discrete uniform distribution assigns equal probability to each outcome in a finite set. It represents fair dice rolls, random card suit selections, or any scenario where all outcomes are equally likely.

$P(X=x)=\frac{1}{n}$

Where n is the total number of possible outcomes.

How do you calculate expected value and variance?

The expected value (mean) and variance are two fundamental measures that summarize a discrete probability distribution.

Expected value

The expected value, denoted μ or E(X), represents the weighted average of all possible values of a discrete random variable. The mean of a discrete probability distribution gives the weighted average of all possible values of the discrete random variable

$E(X)=\mu =\sum _{x}x\cdot P(X=x)$

Each possible value x is multiplied by its probability, and these products are summed across all outcomes.

Variance

The variance, denoted σ² or Var(X), measures the spread of the distribution around the mean. Variance of a discrete probability distribution is defined as the product of the squared difference between each value and the mean, weighted by the PMF.

 $\text{Var}(X)={\sigma }^2=\sum _x(x-\mu {)}^2\cdot P(X=x)$

An alternative computational formula is:

$\text{Var}(X)=E(X^2)-[E(X){]}^2$

Standard deviation

The standard deviation, denoted σ, is the square root of the variance. Standard deviation is a measure of how much each data point varies away from the mean and is often described as the spread of the distribution.