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

(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 \(Y\).