Question

1. Suppose t hat Xhas t he chi-square distribution on p1∈(0, ∞) degrees of f reedom and that, i ndependently, Y has t he chi-square distribution on p2∈(0, p1) degrees of f ree-dom. a. Use moment generating functions to find the distribution of X + Y . b. A naive guess might be that the distribution of X − Y is chi-square on p1− p2 degrees of freedom. Prove that such a guess is wrong by demonstrating that P (X − Y < 0) > 0, whereas a chi-square random variable cannot take on negative values. (i) First show that P(X < 1, Y > 1) > 0. You may quote without proof the well-known fact that the integral of a nonnegative continuous function over an interval of nonzero length can only equal zero if the function itself is identically equal to zero. (ii) Then show that P (X < 1, Y > 1) > 0 implies P (X − Y < 0) > 0. c. Use the one-to-one bivariate transformation formula to find the joint distribution of U := X + Y and V := X − Y .

Answer 1