# Probability Distribution Functions: PMF, PDF, CDF, and Common Distributions

Source: https://www.techinterview.org/post/3233459639/probability-distribution-function/
Updated: 2026-04-27 · techinterview.org

## Probability Distribution Functions: PMF, PDF, CDF, and the Common Distributions

Understanding probability distribution functions is foundational for quant trading interviews, ML / statistics interviews, and any role involving uncertainty modeling. The standard objects — PMF, PDF, CDF, expectation, variance — appear constantly. This guide covers the definitions, the relationships between them, and the standard distributions (normal, exponential, Poisson, geometric, binomial) that interviewers expect candidates to recognize and reason about.

## Probability Mass Function (PMF) — Discrete

For a discrete random variable X taking values in a countable set, the PMF gives the probability of each outcome:

**PMF: p(x) = P(X = x)**

Properties:

- p(x) ≥ 0 for all x

- Σ p(x) = 1 (sum over all possible values)

Example: a fair die has PMF p(x) = 1/6 for x ∈ {1, 2, 3, 4, 5, 6}.

## Probability Density Function (PDF) — Continuous

For a continuous random variable, single points have probability 0; instead, the PDF gives a density:

**PDF: f(x), where P(a ≤ X ≤ b) = ∫[a..b] f(x) dx**

Properties:

- f(x) ≥ 0 for all x

- ∫[-∞..∞] f(x) dx = 1

- f(x) is NOT a probability — it can exceed 1

Example: Standard normal has PDF f(x) = (1/√(2π)) × exp(-x²/2). At x = 0, f(0) ≈ 0.399 — a density, not a probability.

## Cumulative Distribution Function (CDF)

For both discrete and continuous random variables, the CDF gives the probability that X is at most some value:

**CDF: F(x) = P(X ≤ x)**

For discrete: F(x) = Σ p(y) for all y ≤ x.

For continuous: F(x) = ∫[-∞..x] f(t) dt.

Properties:

- F is non-decreasing

- F(-∞) = 0, F(∞) = 1

- F is right-continuous

The CDF is universal — works for any random variable, discrete or continuous, mixed or singular.

## Expectation and Variance

**Expectation (mean):**

- Discrete: E[X] = Σ x × p(x)

- Continuous: E[X] = ∫ x × f(x) dx

**Variance:**

Var(X) = E[(X - E[X])²] = E[X²] - E[X]²

**Standard deviation:** σ = √Var(X), in the same units as X.

## Common Distributions to Know

### Bernoulli

Single trial with success probability p. PMF: p(1) = p, p(0) = 1-p. E[X] = p; Var(X) = p(1-p).

Use case: Coin flip, success/failure events.

### Binomial

Sum of n independent Bernoulli trials. PMF: p(k) = C(n, k) × p^k × (1-p)^(n-k). E[X] = np; Var(X) = np(1-p).

Use case: Number of heads in n flips, conversion counts in A/B tests.

### Geometric

Number of Bernoulli trials until first success. PMF: p(k) = (1-p)^(k-1) × p for k = 1, 2, .... E[X] = 1/p; Var(X) = (1-p)/p².

Use case: First failure after k successes, gambler's ruin variant.

### Poisson

Number of events in fixed interval, rate λ. PMF: p(k) = λ^k × e^(-λ) / k!. E[X] = λ; Var(X) = λ.

Use case: Arrivals in queueing models, rare events, network packet counts.

### Uniform (Continuous)

Equal density on [a, b]. PDF: f(x) = 1/(b-a) for a ≤ x ≤ b, else 0. E[X] = (a+b)/2; Var(X) = (b-a)²/12.

Use case: rand() in most languages, baseline for sampling.

### Exponential

Time until first arrival in a Poisson process, rate λ. PDF: f(x) = λ × e^(-λx) for x ≥ 0. E[X] = 1/λ; Var(X) = 1/λ².

Use case: Time-until-event modeling, memoryless processes (network packet arrival, radioactive decay).

### Normal (Gaussian)

Bell curve. PDF: f(x) = (1/(σ√(2π))) × exp(-(x-μ)²/(2σ²)). E[X] = μ; Var(X) = σ².

Use case: Most aggregate statistics (Central Limit Theorem), measurement errors, asset returns at long horizons.

### Lognormal

X is lognormal if ln(X) is normal. Skewed right; common for asset prices, income distributions.

## Memorylessness Property

The geometric and exponential distributions are *memoryless*: P(X > s + t | X > s) = P(X > t). The waiting time doesn't depend on how long you've already waited.

This is unique to these two distributions. In real-world processes, memorylessness is rarely true exactly, but exponential is often a good approximation for short-horizon events.

## Common Interview Problems

### Computing expectation

"X is uniform on [0, 1]. What's E[X²]?" Answer: ∫₀¹ x² dx = 1/3. Strong candidates do this without paper.

### Variance under transformation

"If Var(X) = σ², what's Var(aX + b)?" Answer: a²σ². Variance is invariant to constant shift; scales by the square of multiplier.

### Sum of independent random variables

"X and Y are independent. Var(X + Y) = ?" Answer: Var(X) + Var(Y). Without independence, you'd need covariance terms.

### Recognize the distribution

"You count k events in time t, where each event is independent. What distribution is this?" Poisson with λ = expected count.

### Compute a tail probability

"X is exponential with rate 1. What's P(X > 2)?" Answer: e^(-2) ≈ 0.135. Use the survival function (1 - CDF).

## Common Mistakes

- **Treating PDF values as probabilities.** f(x) is a density, not a probability. P(X = x) = 0 for any single point in a continuous distribution.

- **Forgetting the area-under-curve = 1 constraint.** A function isn't a valid PDF unless it integrates to 1.

- **Confusing variance and standard deviation.** SD = √Var. SD is in the original units; variance is squared.

- **Misapplying independence.** Sum of variances applies only for independent random variables. With covariance, Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y).

- **Mixing discrete and continuous.** Interval [a, b] for continuous includes both endpoints; for discrete, you need to specify whether endpoints count. The CDF unifies them, but PMF/PDF behavior differs.

## Frequently Asked Questions

### What's the practical difference between PMF and PDF?

  PMF gives a probability at each outcome (discrete). PDF gives a density (continuous; integrate to get probability over an interval). The CDF unifies them. For interviews: know which type the problem implies, and use the right formulas.

### Why is the normal distribution so important?

  Central Limit Theorem: sums of many independent random variables converge to normal. This makes normal a good approximation for aggregate statistics across diverse domains. In quant finance specifically, normality is assumed in many models (Black-Scholes, mean-variance portfolio optimization) even when returns are leptokurtic in practice.

### What does memorylessness mean and why does it matter?

  Future behavior doesn't depend on past history (given a current state). Exponential and geometric have this property; almost no other distributions do. Important in queueing theory and Markov chain analysis where memorylessness simplifies the math considerably.

### How do I quickly compute moments?

  For known distributions, memorize the formulas (E[X] and Var(X)). For functions of random variables, use moment-generating functions (MGFs) or characteristic functions when the algebra gets ugly. Most interview problems can be solved with basic E[X] and Var(X) formulas plus linearity of expectation.

### What books should I use for distribution prep?

  Sheldon Ross's *Introduction to Probability Models* is the classical text. For quant-flavored, Zhou's "green book" (*A Practical Guide to Quantitative Finance Interviews*) has many distribution-related problems. Stat 110 by Joe Blitzstein (Harvard, free online lectures) covers the same material at lecture pace. Most candidates over-prepare here; familiarity with the basic distributions and how to compute moments is sufficient.

See also: [Expected Value](/post/3233474559/expected-value-fair-game-quant-interviews/) • [Conditional Probability](/post/3233474568/conditional-probability-bayesian-quant-interviews/) • [Random Walks and Stopping Times](/post/3233474561/random-walks-stopping-times-quant-interviews/)