The chi-square distribution is defined as the distribution of
Y = X_1**2 + X_2**2 + ... + X_k**2
with X_i independent standard normally distributed (mean 0, std 1).
The expected value of a sum of variables is
E(X_1 + X_2) = E(X_1) + E(X_2)
The expected value of a product of independent...