• Overview
• Details
• Exercises

Discrete random variables map outcomes to numbers. Key families—Bernoulli, Binomial, Geometric, and Poisson—model success/failure trials and event counts. We summarize PMFs, expectations, variances, and relationships.

• Bernoulli(p): PMF P(X=1)=p, P(X=0)=1-p; E[X]=p, Var[X]=p(1-p).
• Binomial(n,p): Sum of n i.i.d. Bernoulli(p). PMF \(\binom{n}{k} p^k (1-p)^{n-k}\). E[X]=np, Var[X]=np(1-p).
• Geometric(p) (trials until first success, support 1,2,...): PMF \((1-p)^{k-1}p\). E[X]=1/p, Var[X]=(1-p)/p^2. Memoryless.
• Poisson(λ): Counts of rare events. PMF \(e^{-\lambda}\lambda^k/k!\). E[X]=Var[X]=\lambda. Sum of independent Poissons is Poisson with rate sum.
• Links: Binomial(n,p) → Poisson(λ=np) as n→∞, p→0 with λ fixed (Poisson limit theorem).
• MGFs/PGFs: Useful for sums and deriving moments.
• Indicators: For event A, I_A has E[I_A]=P(A); summing indicators helps count expectations.

1. Derive the Binomial PMF via counting arguments and confirm it sums to 1.
2. Show the memoryless property of the Geometric distribution.
3. Prove that the sum of independent Poisson(λ_i) variables is Poisson(∑λ_i).
4. Use indicator variables to compute the expected number of heads in n coin flips.