15.6 Exercises

1. Let $X$ have density given by

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

(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.)

(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

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.]