Stat 5101 First Midterm ExamOctober 27, 1999

The exam is closed book and closed notes. You may use a calculator, but shouldn't need to. Put all of your work on this test form (use the back if necessary). Show your work or give an explanation of your answer. No credit for numbers with no indication of where they came from.

1.

Suppose X and Z are independent and identically distributed random variables having mean $\mu$ and variance $\sigma^2$. Define Y = X + Z. What is the correlation of X and Y?

2.

A test for HIV infection has a 4% false positive rate and a 2% false negative rate (that is, there is a 4% chance of a positive test result for a person not actually infected with HIV and a 2% chance of a negative test result for a person actually infected with HIV). Suppose this test is used to screen blood donors, 3% of whom are actually infected with HIV. Given that an individual tests positive, what is the probability that the person is actually infected with HIV?

3.

A random variable has p. d. f. f defined by

$$f(x) = \tfrac{3}{4} (1 - x^2), \qquad -1 < x < +1$$

(and zero elsewhere).

1. Find the mean of X.
2. Find the median of X.
3. Find the c. d. f. of X.

4.

Suppose a random variable has p. d. f. f defined by

$$f(x) = \tfrac{1}{2} x^2 e^{- x}, \qquad 0 < x < + \infty.$$

Find the p. d. f. of the random variable $Y = \sqrt{X}$.

5.

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

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

Find the conditional expectation E(Y | X).

Hint: You may find the following integral formula helpful

$$\int_0^\infty x^n e^{-x} \, d x = n !$$