16.5 Exercises

1. Let $X$ have density $f_X(x) = 2x$ for $0 < x < 1$. Find the density of

(a) $5X - 3$

(b) $4X^3$

(c) $X/(1+X)$

2. Let $X$ have density $f(x) = 2x^{-3}e^{-x^{-2}}$ on the positive real numbers. Find the density of $X^4$.

3. For a fixed $\alpha > 0$ let $X$ have the Pareto density given by

Find the density of $\log(X)$. Recognize this as one of the famous ones and provide its name and parameters.

4. Let $X$ be a random variable. Find the density of $X^2$ if $X$ has the uniform distribution on $(-1, 2)$.

5. Let $U$ have the uniform $(0, 1)$ distribution. For $\lambda > 0$, find a function of $U$ that has the exponential $(\lambda)$ distribution.

6. Let $Z$ be standard normal.

(a) Use the change of variable formula to find the density of $1/Z$. Why do you not have to worry about the event $Z = 0$?

(b) A student who doesn’t like the change of variable formula decides to first find the cdf of $1/Z$ and then differentiate it to get the density. That’s a fine plan. The student starts out by writing $P(1/Z < x) = P(1/x < Z)$ and immediately the course staff say, “Are you sure?” What is the problem with what the student wrote?

(c) For all $x$, find $P(1/Z < x)$.

(d) Check by differentiation that your answer to (c) is consistent with your answer to (a).

7. Let the random variable $X$ have cdf $F$ and let the random variable $Y$ have cdf $G$. You can assume that both $F$ and $G$ are continuous and increasing.

(a) Find a function $h$ such that the random variable $h(X)$ has the uniform $(0, 1)$ distribution.

(b) Use Part (a) to find a function $g$ such that the random variable $g(X)$ has the same distribution as $Y$.