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

$f(x) ~ = ~ \frac{\alpha}{x^{\alpha+1}}, ~~ x > 1$

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 $$P(1/Z < x)$$.

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 $$Y$$.