15.3: Sum of Two Random Processes

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Let X and Y be the random variables associated with the numbers shown on two dice (see above), and Z = X + Y . With P(X = x), P(Y = y), and P(Z = z) being the probabilities associated with the random variables taking specific values x, y or z. Given z = x + y, the event Z = z is the union of the independent events X = k and Y = z − k. We can therefore write

\[P(Z=z)=\sum_{k=-\infty }^{\infty }P(X=k)P(Y=z-k)\]

which is the exact definition of a convolution, also written as

\[P(Z)=P(X)\star P(Y)\]

Numerically calculating the convolution always works, and can be done analytically for some probability distributions. Conveniently, the convolution of two Gaussian distributions is again a Gaussian distribution with a variance that corresponds to the sum of the variances of the individual Gaussians.