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"## Exercises ##"
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"**1.** Let $X$ have density $f_X(x) = 2x$ for $0 < x < 1$. Find the density of\n",
"\n",
"(a) $5X - 3$\n",
"\n",
"(b) $4X^3$\n",
"\n",
"(c) $X/(1+X)$"
]
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"**2.** Let $X$ have density $f(x) = 2x^{-3}e^{-x^{-2}}$ on the positive real numbers. Find the density of $X^4$."
]
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"**3.** For a fixed $\\alpha > 0$ let $X$ have the Pareto density given by\n",
"\n",
"$$f(x) ~ = ~ \\frac{\\alpha}{x^{\\alpha+1}}, ~~ x > 1$$\n",
"\n",
"Find the density of $\\log(X)$. Recognize this as one of the famous ones and provide its name and parameters."
]
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"**4.** Let $X$ be a random variable. Find the density of $X^2$ if $X$ has the\n",
"uniform distribution on $(-1, 2)$."
]
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"**5.** Let $U$ have the uniform $(0, 1)$ distribution. For $\\lambda > 0$, find a function of $U$ that has the exponential $(\\lambda)$ distribution."
]
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"**6.** Let $Z$ be standard normal.\n",
"\n",
"(a) Use the change of variable formula to find the density of $1/Z$. Why do you not have to worry about the event $Z = 0$?\n",
"\n",
"(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?\n",
"\n",
"(c) For all $x$, find $P(1/Z < x)$.\n",
"\n",
"(d) Check by differentiation that your answer to (c) is consistent with your answer to (a)."
]
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"**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.\n",
"\n",
"(a) Find a function $h$ such that the random variable $h(X)$ has the uniform $(0, 1)$ distribution.\n",
"\n",
"(b) Use Part (a) to find a function $g$ such that the random variable $g(X)$ has the same distribution as $Y$."
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