Stat 5131 Second Midterm Exam, November 12, 1997

Problem 1   A random variable X has p. d. f. f defined by

$$f(x) = \begin{cases} \tfrac{5}{8} (1 - x^4), & - 1 < x < + 1 \\ 0, & \text{otherwise} \end{cases}$$

1. Find the mean of X.
2. Find the variance of X.

Problem 2   A pair of random variables X and Y have joint p. d. f.

$$f(x, y) = \tfrac{6}{7} (x + y)^2, \qquad 0 < x < 1,\ 0 < y < 1$$

Find the conditional p. d. f. of X given Y.

Problem 3   The conditional p. d. f. of Y given X is

$$f(y \vert x) = \frac{y + x}{1 + x} e^{-y}, \qquad \text{$y > 0$, $x > 0$}.$$

Find the regression function of Y on X. (You may use the formula $\int_0^\infty x^n e^{-x} \, d x = n !$mentioned in class without rederiving it).

Problem 4   Suppose X, Y, and Z are exchangeable with mean 0, variance 1, and correlation - 0.5.

1. Find $\mathop{\rm var}\nolimits(X + Y)$.
2. Find $\mathop{\rm var}\nolimits(X + Y + Z)$.

Problem 5   For 0 < p < 1, the function

$$\psi(t) = \frac{1 - p}{1 - p e^t}$$

is the moment generating function of a probability distribution. Find the mean and variance.