# 15.6. Exercises¶

1. Let $$X$$ have density given by

$\begin{split}f(x) ~ = ~ \begin{cases} c(1 - x^2), ~~ -1 < x < 1 \\ 0 ~~~~~ \text{otherwise} \end{cases}\end{split}$

Find

(a) $$c$$

(b) the cdf of $$X$$

(c) $$P(\vert X \vert > 0.5)$$

(d) $$E(X)$$

(e) $$SD(X)$$

2. A Zoom call starts at 10:00. Suppose I join the call at a time uniformly distributed over the interval 10:00 to 10:05, and you join the call at an independent time normally distributed with mean 10:03 and SD $$0.5$$ minutes. What is the chance that we both miss the first two minutes of the call?

3. Let $$U_1, U_2, \ldots, U_n$$ be i.i.d. uniform on the interval $$(-1, 1)$$. Assume $$n$$ is large.

(a) Let $$M_n = \min(U_1, U_2, \ldots, U_n)$$. For $$-1 < x < 1$$, find or approximate $$P(M_n > x)$$.

(b) Let $$A_n = \frac{1}{n}\sum_{i=1}^n U_i$$. For $$-1 < x < 1$$, find or approximate $$P(A_n > x)$$.

4. In each of Parts (a) and (b), find $$P(X > 4E(X))$$ if you can. If you can’t, then approximate it if you can, and if you can’t approximate it, then provide the best upper and lower bounds that you can find.

(a) $$X$$ is a non-negative random variable

(b) $$X$$ has the exponential $$(\lambda)$$ distribution

5. Let $$X$$ have the exponential $$(\lambda)$$ distribution and let $$c$$ be a positive constant. Let $$Y=cX$$. Find the survival function of $$Y$$ and hence identify the distribution of $$Y$$.

6. For $$i = 1, 2, \ldots, n$$, let $$X_i$$ have the exponential $$(\lambda_i)$$ distribution, and assume that $$X_1, X_2, \ldots, X_n$$ are independent.

Let $$M = \min(X_1, X_2, \ldots, X_n)$$. Find the distribution of $$M$$. Recognize it as one of the famous ones and provide its name and parameters.

7. For fixed $$\alpha > 2$$, a random variable $$T$$ has the Pareto distribution with shape parameter $$\alpha$$ and possible values $$(1, \infty)$$ if the density of $$T$$ is given by

$f_T(t) ~ = ~ c t^{-(\alpha + 1)}, ~~~ t > 1$

(a) Find $$c$$.

(b) Find the cdf of $$T$$.

(c) Find $$E(T)$$.

(d) Find $$Var(T)$$.

8. Let $$X$$ be a random variable describing the relative change of Bitcoin in a year: a $$1$$ dollar investment in bitcoin will be worth $$X$$ dollars at the end of the year. Jason buys $$100$$ dollars worth of bitcoin. Let $$Y$$ be the profit made on this investment at the end of the year. For example, if $$X=0.9$$, then the profit is $$-10$$ dollars, and if $$X=1.1$$, the profit is $$10$$ dollars.

Assume that $$X$$ has the density, expectation, and variance given below. (This is a lognormal distribution, which you will encounter later in the course.)

$f_X(x) = \frac{1}{x\sqrt{2\pi}} e^{-\frac{(\ln x)^2}{2}}, ~~ x > 0 ~~~~~~~~~~~E(X) = 1 ~~~~~~~~~~~ Var(X) = e^2-e$

(a) Let $$F_X$$ be the cdf of $$X$$ and $$F_Y$$ the cdf of $$Y$$. Without doing any integrals, write $$F_Y$$ in terms of $$F_X$$.

(b) Use Part (a) to find $$f_Y$$, the density of $$Y$$.

(c) Without doing any integrals, find $$E(Y)$$ and $$SD(Y)$$.

9. Let $$X$$ have the bilateral exponential density given by

$f_X(x) ~ = ~ \frac{1}{2}e^{-|x|}, ~~~~ -\infty < x < \infty$

Use what you know about the exponential density to find, without integration,

(a) the cdf of $$X$$

(b) $$E(X)$$

(c) $$Var(X)$$

10. Let $$T$$ have the exponential $$(\lambda)$$ distribution and let $$X$$ be the integer part of $$T$$. Find the distribution of $$X$$. Identify it as one of the famous ones and find its name and parameters.

11. Let $$T$$ be a non-negative random variable that has density $$f$$. For each $$t > 0$$, let $$S(t) = P(T > t)$$. Show that $$E(T) = \int_0^\infty S(t)dt$$.

[Write $$S(t)$$ as an integral of $$f$$, then switch the order of integration.]