Discrete Probability Distributions

Expected value, mathematical expectation, and common discrete distributions: Binomial, Poisson, Geometric, Negative Binomial, and Hypergeometric.
When analyzing data, we often deal with variables whose outcomes are determined by chance. A random variable is a numerical description of the outcome of an experiment. A discrete random variable can take on a countable number of distinct values (e.g., the number of potholes on a 10km stretch of road, or the number of defective bricks in a pallet).

Probability Mass Functions and Mathematical Expectation

The foundational math behind discrete random variables.

Probability Mass Function (PMF), f(x)f(x) or P(X=x)P(X=x)

A function that assigns a probability to each possible value of a discrete random variable. It must satisfy two conditions:
  • f(x)≥0f(x) \ge 0 for all xx.
  • ∑xf(x)=1\sum_{x} f(x) = 1.

Cumulative Distribution Function (CDF), F(x)F(x)

The probability that the random variable XX will take a value less than or equal to xx.
F(x)=P(X≤x)=∑t≤xf(t) F(x) = P(X \le x) = \sum_{t \le x} f(t)

Mathematical Expectation

The expected value represents the theoretical mean of the random variable.

Expected Value (Mean), μ\mu or E[X]E[X]

The long-run average value of the random variable over infinitely many trials. It is the center of the probability distribution.
μ=E[X]=∑xx⋅f(x) \mu = E[X] = \sum_{x} x \cdot f(x)

Variance, σ2\sigma^2 or V(X)V(X)

A measure of the dispersion or spread of the probability distribution around the mean.
σ2=E[(X−μ)2]=∑x(x−μ)2⋅f(x) \sigma^2 = E[(X - \mu)^2] = \sum_{x} (x - \mu)^2 \cdot f(x)
Alternatively, it can be calculated more easily using the computational formula:
σ2=E[X2]−(E[X])2 \sigma^2 = E[X^2] - (E[X])^2

Common Discrete Distributions in Engineering

Specific models used to describe common engineering scenarios.

The Binomial Distribution

Models the number of successes in a fixed number of independent trials.

Binomial Distribution

Applicable when:
  • There are a fixed number of trials (nn).
  • Each trial has only two possible outcomes (Success or Failure).
  • The probability of success (pp) remains constant for each trial.
  • The trials are mutually independent.
The probability of exactly xx successes in nn trials is:
P(X=x)=(nx)px(1−p)n−xfor x=0,1,…,n P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \quad \text{for } x = 0, 1, \dots, n
  • Mean: μ=np\mu = np
  • Variance: σ2=np(1−p)\sigma^2 = np(1-p)

The Poisson Distribution

Models the number of events occurring in a fixed interval of time or space.

Poisson Distribution

Used for rare events where the exact number of trials nn is effectively infinite and pp is very small, but the average rate of occurrence (λ\lambda) is known. Examples include the number of traffic accidents per month at a given intersection, or the number of flaws in a 100m reel of fiber optic cable.
The probability of exactly xx events occurring in a given interval is:
P(X=x)=λxe−λx!for x=0,1,2,… P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!} \quad \text{for } x = 0, 1, 2, \dots
  • Mean: μ=λ\mu = \lambda
  • Variance: σ2=λ\sigma^2 = \lambda
Note: A unique property of the Poisson distribution is that its mean equals its variance.

The Geometric and Negative Binomial Distributions

Models the number of trials needed to achieve a specific number of successes.

Geometric Distribution

Models the number of independent trials XX needed to get the first success. (e.g., How many times must we test a newly designed joint until we observe the first failure, assuming a constant failure probability pp?)
P(X=x)=(1−p)x−1pfor x=1,2,3,… P(X = x) = (1-p)^{x-1}p \quad \text{for } x = 1, 2, 3, \dots
  • Mean: μ=1p\mu = \frac{1}{p}
  • Variance: σ2=1−pp2\sigma^2 = \frac{1-p}{p^2}

Negative Binomial Distribution

A generalization of the geometric distribution. It models the number of independent trials XX needed to get exactly rr successes.
P(X=x)=(x−1r−1)pr(1−p)x−rfor x=r,r+1,… P(X = x) = \binom{x-1}{r-1} p^r (1-p)^{x-r} \quad \text{for } x = r, r+1, \dots

The Hypergeometric Distribution

Models sampling without replacement.

Hypergeometric Distribution

Unlike the Binomial distribution where pp is constant (sampling with replacement), the Hypergeometric distribution is used when sampling without replacement from a finite population of size NN, containing exactly KK successes. (e.g., Selecting 5 concrete cylinders from a batch of 50, where 3 are known to be defective).
P(X=x)=(Kx)(N−Kn−x)(Nn) P(X = x) = \frac{\binom{K}{x} \binom{N-K}{n-x}}{\binom{N}{n}}
  • Mean: μ=n(KN)\mu = n \left(\frac{K}{N}\right)
Key Takeaways
  • Random Variables: Numerical values assigned to experimental outcomes.
  • Expected Value (E[X]E[X]): The long-run average of a discrete distribution.
  • Binomial: Used for independent trials with exactly two outcomes (success/failure) and constant probability pp.
  • Poisson: Used for modeling the number of rare events occurring within a continuous interval (time, area, volume).
  • Geometric/Negative Binomial: Focuses on the number of trials needed to achieve a specified number of successes.
  • Hypergeometric: Used for finite populations when sampling without replacement (probability changes trial-to-trial).
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