Discrete Probability Distributions

Discrete Probability Distributions

$Learning Objectives$

Understand the concepts of probability mass functions (PMF) and cumulative distribution functions (CDF).
Calculate the expected value (mean) and variance of a discrete random variable.
Apply the Binomial and Poisson distributions to model relevant civil engineering scenarios.
Apply the Geometric and Negative Binomial distributions for scenarios involving trials to achieve successes.
Use the Hypergeometric distribution to model probabilities when sampling without replacement from finite batches.

Discrete probability distributions are foundational for modeling scenarios where outcomes are countable, such as the number of structural defects, rainfall events, or equipment failures. This lesson covers key distributions including Binomial, Poisson, Geometric, Negative Binomial, and Hypergeometric, and how to apply them to solve practical engineering problems.

$Introduction to Random Variables$

When analyzing engineering 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, the number of defective bricks in a pallet, or the number of extreme weather events in a year). Understanding these variables allows engineers to quantify risk and predict the frequency of specific events.

Probability Mass Functions and Mathematical Expectation

$Probability Functions and Expectation$

The foundational math behind discrete random variables.

$f(x)$

A function that assigns a probability to each possible value of a discrete random variable.

$PMF Conditions$

$f(x) \ge 0$ for all $x$ .
$\sum_{x} f(x) = 1$ .

$F(x)$

The probability that the random variable $X$ will take a value less than or equal to $x$ .

Cumulative Distribution Function

Formula for calculating the CDF of a discrete random variable.

F(x) = P(X \le x) = \sum_{t \le x} f(t)

Variables

Symbol	Description	Unit
$F(x)$	Cumulative Distribution Function	-
$X$	Discrete random variable	-
$x$	Specific value of the random variable	-
$f(t)$	Probability Mass Function evaluated at t	-

Mathematical Expectation

$Mathematical Expectation and Variance$

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

$\mu$

The long-run average value of the random variable over infinitely many trials. It is the center of the probability distribution.

Expected Value (Mean)

Formula for calculating the expected value.

\mu = E[X] = \sum_{x} x \cdot f(x)

Variables

Symbol	Description	Unit
$\mu$	Expected value or mean	-
$E[X]$	Expected value of random variable X	-
$x$	Specific value of the random variable	-
$f(x)$	Probability Mass Function evaluated at x	-

$\sigma^2$

A measure of the dispersion or spread of the probability distribution around the mean.

Variance

Formula for calculating the variance.

\sigma^2 = E[(X - \mu)^2] = \sum_{x} (x - \mu)^2 \cdot f(x)

Variables

Symbol	Description	Unit
$\sigma^2$	Variance	-
$E$	Expected value operator	-
$X$	Discrete random variable	-
$\mu$	Expected value or mean	-
$x$	Specific value of the random variable	-
$f(x)$	Probability Mass Function evaluated at x	-

Computational Formula for Variance

An alternative, easier formula for calculating variance.

\sigma^2 = E[X^2] - (E[X])^2

Variables

Symbol	Description	Unit
$\sigma^2$	Variance	-
$E[X^2]$	Expected value of the square of X	-
$E[X]$	Expected value of X	-

Common Discrete Distributions in Engineering

$Common Discrete Distributions$

Specific models used to describe common engineering scenarios.

The Binomial Distribution

$Introduction to the Binomial Distribution$

The Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials with the same success probability. In civil engineering, an example is the number of independently selected concrete cylinders that exceed a specified strength threshold in a fixed sample. Repeated days affected by weather should not be treated as independent trials unless that assumption is justified.

$Binomial Distribution$

A distribution that models the probability of exactly $x$ successes in $n$ independent trials.

$Binomial Distribution Conditions$

There are a fixed number of trials ( $n$ ).
Each trial has only two possible outcomes (Success or Failure).
The probability of success ( $p$ ) remains constant for each trial.
The trials are mutually independent.

Binomial Distribution Formula

Formula to calculate the probability of exactly x successes in n trials.

P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \quad \text{for } x = 0, 1, \dots, n

Variables

Symbol	Description	Unit
$P(X = x)$	Probability of exactly x successes	-
$n$	Total number of trials	-
$x$	Number of successes	-
$p$	Probability of success on a single trial	-
$(1-p)$	Probability of failure on a single trial	-

Binomial Mean and Variance

Formulas for the mean and variance of a Binomial distribution.

\mu = np \quad \text{and} \quad \sigma^2 = np(1-p)

Variables

Symbol	Description	Unit
$\mu$	Expected value or mean	-
$\sigma^2$	Variance	-
$n$	Total number of trials	-
$p$	Probability of success on a single trial	-

The Poisson Distribution

$Introduction to the Poisson Distribution$

The Poisson distribution models a count over a defined time, length, area, volume, or other exposure when the count process is adequately described by a Poisson model. For a homogeneous Poisson process, the event rate is stable over the stated exposure and counts in disjoint intervals are independent. Rarity alone does not establish that a Poisson model is appropriate.

$Poisson Distribution$

Models the nonnegative integer count $X$ in a specified exposure when its distribution is Poisson with mean count $\lambda$ . Under a homogeneous Poisson process with rate $r$ per unit exposure $E$ , $\lambda=rE$ . This is a count model with stated assumptions, not a rule that every rare event follows a Poisson distribution. Separately, a Poisson distribution may approximate a Binomial count when the fixed-trial model and approximation accuracy support it.

Poisson Distribution Formula

Formula to calculate the probability of exactly x events occurring in a given interval.

P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!} \quad \text{for } x = 0, 1, 2, \dots

Variables

Symbol	Description	Unit
$P(X = x)$	Probability of exactly x events	-
$\lambda$	Expected event count over the specified exposure	-
$e$	Euler's number (approximately 2.71828)	-
$x$	Number of events	-

A stable event rate r over exposure E and twice that exposure, with expected counts λ=rE and 2λ=r(2E). — Poisson Counts Scale with Exposure
Analytical schematic of a homogeneous Poisson count model: for a stable rate r per unit exposure, the expected count over exposure E is λ=rE. Event marks are illustrative locations, not measured data, and an individual count is random.

Poisson Mean and Variance

Formulas for the mean and variance of a Poisson distribution.

\mu = \lambda \quad \text{and} \quad \sigma^2 = \lambda

Variables

Symbol	Description	Unit
$\mu$	Expected value or mean	-
$\sigma^2$	Variance	-
$\lambda$	Average rate of occurrence in the interval	-

$Unique Property of Poisson Distribution$

A unique property of the Poisson distribution is that its mean equals its variance.

The Geometric and Negative Binomial Distributions

$Geometric and Negative Binomial Models$

The Geometric and Negative Binomial distributions model the number of trials needed to achieve a specific number of successes. For instance, testing identical soil samples until the first one fails a compaction test (Geometric), or testing until exactly three samples fail (Negative Binomial).

$Geometric Distribution$

Models the number of independent trials $X$ 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 $p$ ?)

Geometric Distribution Formula

Formula for calculating the probability of getting the first success on the x-th trial.

P(X = x) = (1-p)^{x-1}p \quad \text{for } x = 1, 2, 3, \dots

Variables

Symbol	Description	Unit
$P(X = x)$	Probability of first success on the x-th trial	-
$p$	Probability of success on a single trial	-
$x$	Trial number on which the first success occurs	-

Geometric Mean and Variance

Formulas for calculating the mean and variance of a geometric distribution.

\mu = \frac{1}{p} \quad \text{and} \quad \sigma^2 = \frac{1-p}{p^2}

Variables

Symbol	Description	Unit
$\mu$	Expected value or mean	-
$\sigma^2$	Variance	-
$p$	Probability of success on a single trial	-

$Negative Binomial Distribution$

A generalization of the geometric distribution. It models the number of independent trials $X$ needed to get exactly $r$ successes.

Negative Binomial Distribution Formula

Formula for calculating the probability of getting the r-th success on the x-th trial.

P(X = x) = \binom{x-1}{r-1} p^r (1-p)^{x-r} \quad \text{for } x = r, r+1, \dots

Variables

Symbol	Description	Unit
$P(X = x)$	Probability of r-th success on the x-th trial	-
$x$	Trial number on which the r-th success occurs	-
$r$	Total number of successes desired	-
$p$	Probability of success on a single trial	-

Negative Binomial Mean and Variance

Formulas for calculating the mean and variance of a negative binomial distribution.

\mu = \frac{r}{p} \quad \text{and} \quad \sigma^2 = \frac{r(1-p)}{p^2}

Variables

Symbol	Description	Unit
$\mu$	Expected value or mean	-
$\sigma^2$	Variance	-
$r$	Total number of successes desired	-
$p$	Probability of success on a single trial	-

The Hypergeometric Distribution

$Sampling Without Replacement$

The Hypergeometric Distribution models sampling without replacement from a finite population. This is critical in quality control where inspecting an item destroys it or removes it from the batch, changing the probability for subsequent selections.

$Hypergeometric Distribution$

Unlike the Binomial distribution where $p$ is constant (sampling with replacement), the Hypergeometric distribution is used when sampling without replacement from a finite population of size $N$ , containing exactly $K$ successes. (e.g., Selecting 5 concrete cylinders from a batch of 50, where 3 are known to be defective).

Hypergeometric Distribution Formula

Formula for calculating hypergeometric probabilities.

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

Variables

Symbol	Description	Unit
$P(X = x)$	Probability of exactly x successes in the sample	-
$N$	Total population size	-
$K$	Total number of successes in the population	-
$n$	Sample size	-
$x$	Number of successes in the sample	-

Hypergeometric Mean and Variance

Formulas for the mean and variance of a hypergeometric distribution.

\mu = n \left(\frac{K}{N}\right) \quad \text{and} \quad \sigma^2 = n \left(\frac{K}{N}\right) \left(\frac{N-K}{N}\right) \left(\frac{N-n}{N-1}\right)

Variables

Symbol	Description	Unit
$\mu$	Expected value or mean	-
$\sigma^2$	Variance	-
$N$	Total population size	-
$K$	Total number of successes in the population	-
$n$	Sample size	-

In an illustrative lot with three defective items among eight, the next-draw defect probability changes after a draw without replacement but stays 3/8 with replacement. — Replacement Changes the Next-Draw Probability
Illustrative finite batch with 3 marked items among 8. With replacement, the next-draw probability remains 3/8; without replacement it becomes 2/7 after a marked item or 3/7 after an unmarked item. These values explain why finite sampling without replacement follows a Hypergeometric model rather than independent Binomial trials.

$Interactive Simulation$

Interact with the simulation below to visualize various discrete probability distributions.

Engineering Data Analysis

Discrete Probability Distributions Explorer

Loading chart...

P(X=x) = \binom{10}{x} (0.50)^x (0.50)^{10-x}

Number of Trials (

n

)10

Success Probability (

p

)0.50

Theoretical Properties

Mean (

\mu

):5.00

Variance (

\sigma^2

):2.50

Std Dev (

\sigma

):1.58

$Interactive Simulation$

Compare the geometric and hypergeometric distributions under different parameters to see how sampling without replacement alters success probabilities.

Engineering Data Analysis • Topic 5

Discrete Probability Distributions Sandbox

Distribution Model

Success Prob (

p

)0.30

Loading chart...

Mean (Expected Value)3.333

Variance7.778

Select a Discrete Count Model

Match the random variable and sampling mechanism to a discrete probability model before calculating probabilities.

Use Hypergeometric for a finite population sampled without replacement, with N, K, and n identified. Use Binomial for a fixed number of independent Bernoulli trials with common p. A Poisson approximation to a Binomial count is a separate option only when its accuracy is adequate for the intended use; no universal rarity cutoff establishes that. A Poisson count over exposure E with mean λ = rE requires a suitable count process; a homogeneous Poisson process has a stable rate r and independent increments. Geometric and Negative Binomial models count independent constant-p trials to the first or r-th success. If the data-generation process is unclear or these conditions are unsupported, hold and establish an appropriate model.

Key Takeaways

Random Variables: Numerical values assigned to experimental outcomes.
Expected Value ( $E[X]$ ): The long-run average of a discrete distribution.
Binomial: Used for independent trials with exactly two outcomes (success/failure) and constant probability $p$ .
Poisson: Models a count over a stated exposure when the Poisson count assumptions are adequate; for a homogeneous process, the expected count is rate times exposure ( $\lambda=rE$ ). A fixed-trial Poisson approximation is a separate modeling choice.
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).

$References$

NIST/SEMATECH e-Handbook: Binomial Distribution — fixed number of independent trials with two outcomes and a common success probability.
NIST/SEMATECH e-Handbook: Poisson Distribution — count model for events in a specified interval.
NIST/SEMATECH e-Handbook: Poisson Approximation for Counts — context-dependent approximation of Binomial counts; its criteria are not universal acceptance thresholds.
NIST Dataplot Reference Manual: Hypergeometric Distribution — finite-population sampling without replacement.

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Learning Objectives

Introduction to Random Variables

Probability Mass Functions and Mathematical Expectation

Probability Functions and Expectation

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

PMF Conditions

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

Cumulative Distribution Function

Mathematical Expectation

Mathematical Expectation and Variance

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

Expected Value (Mean)

Variance, σ2\sigma^2σ2 or V(X)V(X)V(X)

Variance

Computational Formula for Variance

Common Discrete Distributions in Engineering

Common Discrete Distributions

The Binomial Distribution

Introduction to the Binomial Distribution

Binomial Distribution

Binomial Distribution Conditions

Binomial Distribution Formula

Binomial Mean and Variance

The Poisson Distribution

Introduction to the Poisson Distribution

Poisson Distribution

Poisson Distribution Formula

Poisson Mean and Variance

Unique Property of Poisson Distribution

The Geometric and Negative Binomial Distributions

Geometric and Negative Binomial Models

Geometric Distribution

Geometric Distribution Formula

Geometric Mean and Variance

Negative Binomial Distribution

Negative Binomial Distribution Formula

Negative Binomial Mean and Variance

The Hypergeometric Distribution

Sampling Without Replacement

Hypergeometric Distribution

Hypergeometric Distribution Formula

Hypergeometric Mean and Variance

Interactive Simulation

Engineering Data Analysis

Theoretical Properties

Interactive Simulation

Engineering Data Analysis • Topic 5

References

$Learning Objectives$

$Introduction to Random Variables$

$Probability Functions and Expectation$

$f(x)$

$PMF Conditions$

$F(x)$

$Mathematical Expectation and Variance$

$\mu$

$\sigma^2$

$Common Discrete Distributions$

$Introduction to the Binomial Distribution$

$Binomial Distribution$

$Binomial Distribution Conditions$

$Introduction to the Poisson Distribution$

$Poisson Distribution$

$Unique Property of Poisson Distribution$

$Geometric and Negative Binomial Models$

$Geometric Distribution$

$Negative Binomial Distribution$

$Sampling Without Replacement$

$Hypergeometric Distribution$

$Interactive Simulation$

$Interactive Simulation$

$References$