Multivariate Random Variables

A joint PDF or PMF on a region is given, and you are asked for a marginal, a conditional, or a probability over a sub-region.

Sketch the joint support — this is the single biggest accuracy boost on multivariate problems. To get a marginal, integrate (or sum) the joint over the other variable across that support. For a conditional density, divide the joint by the marginal of the conditioning variable. For a probability over a region, set the integration order to match the geometry.

f_{X|Y}(x|y) = f(x, y) / f_Y(y); marginals are f_Y(y) = ∫ f(x, y) dx.

You are given a joint distribution or summary statistics and asked for covariance, correlation, or E[X | Y].

Compute E[XY], E[X], and E[Y] separately, then Cov(X, Y) = E[XY] - E[X]E[Y]. Divide by σ_X σ_Y for correlation. For conditional expectation, work with the conditional density and integrate x · f_{X|Y}(x|y) dx; use the law of total expectation E[X] = E[E[X|Y]] when the conditional is easier to write than the joint.

Cov(X, Y) = E[XY] - E[X]E[Y]; Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X, Y).

Independent variables with named distributions are added, transformed, or combined, and the question asks for the distribution or a probability about the result.

Match to a closure result first: sums of independent normals are normal, sums of independent Poissons are Poisson, sums of independent gammas with common rate are gamma. If no closure rule applies, use convolution for sums or MGFs to identify the resulting distribution. For products and quotients, the CDF method is usually the cleanest.

M_{X + Y}(t) = M_X(t) M_Y(t) for independent X and Y.

A sample size is large enough that you are expected to approximate the distribution of the sum or sample mean with a normal.

Identify the mean and variance of a single observation, multiply by n for the sum mean and variance (or divide by n for the sample mean variance), then standardize and use the standard normal table. Add a continuity correction when the underlying variable is integer-valued.

(X̄ - μ) / (σ / √n) is approximately standard normal for large n.