Convolutions and Sums of Random Variables for Exam P

The Convolution Formula

If X and Y are independent continuous random variables, the PDF of Z = X + Y is the convolution f_Z(z) = integral of f_X(x) * f_Y(z - x) dx. For discrete variables, p_Z(z) = sum over x of p_X(x) * p_Y(z - x). This integral (or sum) can be tedious to compute directly, so alternative methods are often preferred on Exam P.

Example: If X and Y are independent Uniform(0, 1), the convolution gives f_Z(z) = z for 0 < z < 1 and f_Z(z) = 2 - z for 1 < z < 2 (a triangular distribution).

MGF Method for Sums

The moment generating function method is usually faster: M_{X+Y}(t) = M_X(t) * M_Y(t) for independent X, Y. If you can identify the resulting MGF as belonging to a known distribution, you are done. This works perfectly for sums of independent normals, Poissons, gammas (same rate), and binomials (same p).

Example: If X ~ Gamma(3, 2) and Y ~ Gamma(5, 2) are independent, M_{X+Y}(t) = (2/(2-t))^3 * (2/(2-t))^5 = (2/(2-t))^8, so X + Y ~ Gamma(8, 2).

CDF Method for Sums

Sometimes neither convolution nor MGFs are convenient. The CDF method finds P(X + Y <= z) = double integral over the region where x + y <= z. This is especially useful when X and Y are not independent. For Exam P, draw the integration region carefully: typically a triangle or trapezoid. The CDF approach can also handle functions other than sums, such as Z = X * Y or Z = X / Y, by adjusting the region appropriately.