Discrete Probability Distributions

A random variable $X$ is a function from the sample space to the real numbers. Discrete random variables take a countable set of values (integers, fractions). Continuous random variables take any value in an interval.

The probability mass function (PMF) of a discrete $X$ gives $P(X=x_i)$ for each possible value. The PMF must satisfy $P(X=x_i)\geq 0$ and $\sum_i P(X=x_i) = 1$.

The expected value (population mean) is $\mu = E(X) = \sum_i x_i P(X=x_i)$. It is the long-run average over infinitely many independent repetitions. It does not need to be a possible value of $X$ — a fair die has $E(X)=3.5$.

The variance of $X$ is $\sigma^2 = \text{Var}(X) = E[(X-\mu)^2] = E(X^2)-[E(X)]^2$. The standard deviation $\sigma = \sqrt{\text{Var}(X)}$ is in the same units as $X$.

Key properties: $E(aX+b)=aE(X)+b$, $\text{Var}(aX+b) = a^2\text{Var}(X)$. For independent variables: $E(X+Y)=E(X)+E(Y)$ always, and $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)$ only when $X$ and $Y$ are independent.

The binomial distribution models the number of successes $X$ in exactly $n$ independent trials, each with the same success probability $p$. It arises in coin flips, quality control, clinical trials, and polling.

Requirements (BINS): Binary outcomes only, Independent trials, Number of trials $n$ is fixed in advance, Same probability $p$ for each trial.

PMF: $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$, for $k=0,1,\ldots,n$. The $\binom{n}{k}$ term counts the number of ways to arrange exactly $k$ successes among $n$ trials.

Mean: $\mu = np$. Standard deviation: $\sigma = \sqrt{np(1-p)}$. When $p=0.5$ the distribution is symmetric; when $p\neq 0.5$ it is skewed toward $0$ (if $p<0.5$) or toward $n$ (if $p>0.5$).

Cumulative probabilities $P(X\leq k)$ are computed by summing the PMF or using tables/software. For $P(X\geq k)$, use the complement: $1-P(X\leq k-1)$.

The Poisson distribution models the count of independent events in a fixed interval of time, area, or volume, when events occur at a constant average rate $\lambda > 0$.

PMF: $P(X=k) = e^{-\lambda}\lambda^k/k!$, for $k=0,1,2,\ldots$. The Poisson distribution has no upper bound — in principle any count is possible.

Both the mean and variance equal $\lambda$: $E(X)=\text{Var}(X)=\lambda$. A Poisson random variable is characterised entirely by $\lambda$.

Typical applications: calls arriving at a call centre per hour, photons hitting a detector per second, mutations in a DNA sequence per million base pairs, accidents at an intersection per month.

Poisson as a limit of the binomial: when $n$ is large, $p$ is small, and $np=\lambda$ is moderate, the binomial $B(n,p)$ is well-approximated by Poisson($\lambda$). This is the rare-event approximation.

The geometric distribution models the number of trials until the first success. $P(X=k) = (1-p)^{k-1}p$. Mean $= 1/p$. The geometric distribution is memoryless: $P(X>m+n|X>m) = P(X>n)$.

The negative binomial distribution generalises: it counts the number of trials needed to achieve exactly $r$ successes. Mean $= r/p$, variance $= r(1-p)/p^2$.

The hypergeometric distribution models sampling without replacement from a finite population. Used when the population is small enough that each draw changes the probabilities — for example, selecting $5$ cards from a deck without replacing them.

Binomial: fixed $n$ trials, each independent with probability $p$, counting the number of successes.

Poisson: counting rare, independent events in a fixed window; no upper bound on the count; mean and variance are equal.

Geometric: counting the number of trials until the first success; memoryless.

Hypergeometric: like the binomial but sampling without replacement from a finite population; used when the population is small relative to the sample.

Decision guide: if $n$ is large, $p$ is small, and $np$ is moderate, the Poisson approximation to the binomial is accurate and simpler to compute.