Discrete Probability Distributions Reference Cheat Sheet

Key Facts

• A probability mass function satisfies $P(X=x) \ge 0$ for every value $x$ and $\sum_x P(X=x)=1$.
• The cumulative distribution function is $F(x)=P(X \le x)$, so for discrete variables $P(a \le X \le b)=F(b)-F(a-1)$ when values are integers.
• The expected value of a discrete random variable is $E(X)=\sum_x xP(X=x)$.
• The variance is $\operatorname{Var}(X)=E\left(X^2\right)-\left[E(X)\right]^2$, where $E\left(X^2\right)=\sum_x x^2P(X=x)$.
• For a Bernoulli random variable with success probability $p$, $P(X=1)=p$, $P(X=0)=1-p$, $E(X)=p$, and $\operatorname{Var}(X)=p(1-p)$.
• For a binomial random variable $X \sim \operatorname{Bin}(n,p)$, $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$, $E(X)=np$, and $\operatorname{Var}(X)=np(1-p)$.
• For a geometric random variable counting trials until the first success, $P(X=k)=(1-p)^{k-1}p$, $E(X)=\frac{1}{p}$, and $\operatorname{Var}(X)=\frac{1-p}{p^2}$.
• For a Poisson random variable $X \sim \operatorname{Pois}(\lambda)$, $P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}$, $E(X)=\lambda$, and $\operatorname{Var}(X)=\lambda$.

Vocabulary

Discrete random variable

A random variable whose possible values can be listed, such as $0,1,2,3$ or another countable set.

Probability mass function

A function that gives $P(X=x)$ for each possible value $x$ of a discrete random variable.

Cumulative distribution function

A function defined by $F(x)=P(X \le x)$ that gives the probability of being at or below a value.

Expected value

The long-run average value of a random variable, calculated by $E(X)=\sum_x xP(X=x)$.

Variance

A measure of spread around the mean, calculated by $\operatorname{Var}(X)=E\left(X^2\right)-\left[E(X)\right]^2$.

Independent trials

Trials are independent when the outcome of one trial does not change the probabilities for the others.