Suppose that we have d independent standard normal random variables z1, . . , zd, where each zi ∼ n(0, 1). we say that a continuous random variable x follows a chi-squared (χ 2 ) distribution with d degrees of freedom if x d= z 2 1 + · · · + z 2 d . remember that d= means “equal in distribution.” we write this as x ∼ χ 2 d , and it is possible to show that the pdf of x is f(x) = 1 2 d/2γ(d/2) · x d/2−1 · exp(−x/2), where γ(·) is the gamma function, which generalizes the factorial function to continuous numbers (see wikipedia). note: you don’t have to derive this pdf. the chi-squared distribution plays an important role in statistical inference. for now, your job is simply to compute e(x) and var(x). hint: use the relationship between the chi-squared and normal distribution, rather than the pdf directly. this is much easier.