40. ???? and ???? are jointly normally distributed with joint density function given by 1

????(????, ????) =

2????σₓσᵧ√(1 − ????²)

× ????????????

− 1/2 (1 − ????²)[(???? − ????ₓ)²/σₓ² + (???? − ????ᵧ)²/σᵧ²

− 2????(???? − ????ₓ)(???? − ????ᵧ)/(σₓσᵧ)]

(a) Show that the conditional distribution of Y, given X = x, is normal with mean µy + σx (x - µx) and variance σ²(1 - p²).
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(b) Show that Corr(X, Y) = p.

(c) Argue that X and Y are independent if and only if p = 0.