# Question: A joint random variable X1 X2 is said to have

A joint random variable (X1, X2) is said to have a bivariate normal distribution if its joint density is given by

for –∞ < s < ∞ and –∞ < t < ∞.

(a) Show that E(X1) = μX1 and E(X2) = σX2.

(b) Show that variance (X1) = σ2X1, variance (X2) = σ2X2, and the correlation coefficient is

(c) Show that marginal distributions of X1 and X2 are normal.

(d) Show that the conditional distribution of X1, given X2 = x2, is normal with mean

and variance σ2x1(1 – p2).

for –∞ < s < ∞ and –∞ < t < ∞.

(a) Show that E(X1) = μX1 and E(X2) = σX2.

(b) Show that variance (X1) = σ2X1, variance (X2) = σ2X2, and the correlation coefficient is

(c) Show that marginal distributions of X1 and X2 are normal.

(d) Show that the conditional distribution of X1, given X2 = x2, is normal with mean

and variance σ2x1(1 – p2).

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