Univariate Random Variables

You are given a PMF or PDF (possibly piecewise) and asked to find a probability, the CDF, or to verify that the function is a valid distribution.

First confirm the function integrates or sums to one on its support. Then translate "P(X < a)" or "P(a < X < b)" into the corresponding integral or sum, paying attention to whether endpoints are included for discrete versus continuous variables. Build the CDF piecewise so you can answer multiple questions from one expression.

For continuous X, P(X = x) = 0; for discrete X, sum probabilities at each point.

A distribution is provided and you must compute E[X], Var(X), E[g(X)], or extract a moment from a moment generating function.

Apply the definitions: E[X] is the integral or sum of x times the density, Var(X) = E[X^2] - (E[X])^2, and the n-th derivative of M(t) at t = 0 gives E[X^n]. For E[g(X)], plug the transformation into the density without changing variables. Use linearity (E[aX + b] = aE[X] + b, Var(aX + b) = a^2 Var(X)) before reaching for the definition.

Var(X) = E[X^2] - (E[X])^2; M_X(t) = E[e^{tX}] generates moments by differentiation at 0.

A scenario maps cleanly onto a named distribution and you must identify it and compute a probability, mean, or variance.

Match the wording to the canonical setup (trials with two outcomes, time-between-events, max-of-uniforms, etc.) and verify the parameterization the exam is using — gamma in particular has a rate vs scale convention and Pareto has multiple forms. Then apply the known formulas for mean, variance, and tail probability. If the scenario is "given an exponential, find conditional probability past time t," exploit memorylessness.

Exponential and geometric are memoryless; sum of independent gammas with the same rate is gamma; sum of independent normals is normal.

You are given the distribution of X and asked for the distribution, density, or expectation of Y = g(X).

Use the CDF method as the default: write F_Y(y) = P(g(X) ≤ y), invert g to get a statement about X, then differentiate to get the density. For monotone g, the Jacobian shortcut f_Y(y) = f_X(g^{-1}(y)) · |dg^{-1}/dy| is fast. For E[g(X)], you can usually skip finding f_Y entirely and integrate g(x) f_X(x) dx.

f_Y(y) = f_X(g^{-1}(y)) |d g^{-1}(y) / dy| for monotonic g.