X and Y are independent with moment generating functions (1 - 2t)^{-3} and (1 - 2t)^{-5} for t < 1/2. Identify each distribution, give the distribution of X + Y, and compute P(X + Y > 26.30) given that 26.30 is the 95th percentile of a chi-square with 16 degrees of freedom.

X and Y are independent with moment generating functions (1 - 2t)^{-3} and (1 - 2t)^{-5} for t < 1/2. Identify each distribution, give the distribution of X + Y, and compute P(X + Y > 26.30) given that 26.30 is the 95th percentile of a chi-square with 16 degrees of freedom.

Approach: Match each transform against the gamma family's moment generating function, then multiply the two transforms and match the product.

0.05. The moment generating function of a gamma distribution with shape k and scale 2 is (1 - 2t)^{-k} for t < 1/2, so X is Gamma(3, 2) and Y is Gamma(5, 2), which are chi-square variables with 6 and 10 degrees of freedom. Independence makes the transform of the sum the product of the transforms, (1 - 2t)^{-3}*(1 - 2t)^{-5} = (1 - 2t)^{-8}, which is Gamma(8, 2), a chi-square with 16 degrees of freedom. The uniqueness theorem says a moment generating function finite in a neighbourhood of zero determines the distribution, so no further work is needed to identify the sum. With 26.30 given as the 95th percentile the tail probability is 0.05. The same argument shows the chi-square family is closed under addition of independent members, which is why degrees of freedom add across independent sums of squares.

Follow-up: X and Y are each chi-square with 6 and 10 degrees of freedom but are correlated. What can you still say about the mean and variance of X + Y, and what breaks?

Key concepts: moment generating function, chi-square distribution, gamma distribution, uniqueness theorem.