22. The random variables X and Y are said to have a bivariate normal distribution if their joint density function is given by

$$f(x, y) = \frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-ho^2}}$$

$$\times \exp\left[-\frac{1}{2(1-ho^2)}\left[\left(\frac{x-\mu_x}{\sigma_x}\right)^2 + \left(\frac{y-\mu_y}{\sigma_y}\right)^2-2ho\left(\frac{x-\mu_x}{\sigma_x}\right)\left(\frac{y-\mu_y}{\sigma_y}\right)\right]\right]$$

(a) Show that the conditional density of X, given that Y = y, is the normal density with parameters

$$\mu_x + ho\frac{\sigma_x}{\sigma_y}(y-\mu_y)$$ and $$\sigma_x^2(1-ho^2)$$

(b) Show that X and Y are both normal random variables with respect to parameters μ, σ_x^2 and μ, σ_y^2.

(c) Show that X and Y are independent when p = 0.

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