Continuous Random Variables

Let $X$be a random variable that takes on values between$0$ and$c$. That is$P\left\{0\le X\le c\right\}=1$. Show that

$Var\left(X\right)\le \frac{c^2}{4}$

Hint: One approach is to first argue that

$E\left[X^2\right] \le cE\left[X\right]$

and then use this inequality to show that

$Var\left(X\right) \le c^2\left[\alpha \right(1-\alpha \left)\right] where \alpha = \frac{E\left[X\right]}{c}$

Show that$E\left[Y\right]={\int }_0^{\infty }P\left\{Y>y\right\} dy-{\int }_0^{\infty }P\left\{Y<-y\right\} dy$.

Hint: Show that ${\int }_0^{\infty }P\left\{Y<-y\right\}dy=-{\int }_{-\infty }^{0}x{f}_y\left(x\right)dx$

${\int }_0^{\infty }P\left\{Y>y\right\}dy={\int }_0^{\infty }x{f}_y\left(x\right)dx$

Show that if $X$ has density function$f$, then$E\left[g\right(X\left)\right] = {\int }_{-\infty }^{\infty } g\left(x\right)f\left(x\right) dx$

Hint: Using Theoretical Exercise$5.2$, start with$E\left[g\right(X\left)\right]={\int }_0^{\infty }P\{g\left(X\right)>y\}dy-{\int }_0^{\infty }P\{g\left(X\right)<-y\}dy$ and then proceed as in the proof given in the text when$g\left(X\right) \ge 0.$

Prove Corollary$2.1$.

Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g'.

 (a) Show that E[g'(Z)]=E[Zg(Z)]; 

(b) Show that E[Z^n+$1$]=nE[Z^n-$1$].

(c) Find E[Z^$4$]. 

 Use the identity of Theoretical Exercise 5.5 to derive E[X2] when X is an exponential random variable with parameter λ. 

If X is an exponential random variable with parameter λ, and c > 0, show that cX is exponential with parameter λ/c

The median of a continuous random variable having distribution function F is that value m such that F(m) = $\frac{1}{2}$. That is, a random variable is just as likely to be larger than its median as it is to be smaller. Find the median of X if X is 

(a) uniformly distributed over (a, b); 

(b) normal with parameters μ,σ^$2$; 

(c) exponential with rate λ. 

The standard deviation of $X$, denoted $SD\left(X\right)$, is given by

$SD\left(X\right)=\sqrt{Var\left(X\right)}$

Find $SD(a X+b)$ if $X$ has variance ${\sigma }^2$.

The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given by

$f\left(x\right) = \left\{\begin{array}{ll}a{x}^{2 }{e}^{-b{x}^2}& x\ge 0 \\ 0& x<0\end{array}\right.$

where$b = m/2kT$ and$m$ denote, respectively,

Boltzmann’s constant, the absolute temperature of the gas,

and the mass of the molecule. Evaluate $a$in terms of$b$.

Show that $Z$ is a standard normal random variable; then, for,$x>0$

1. $P\{Z>x\}=P\{Z<-x\}$
2. $P\left\{\right|Z|>x\}=2P\{Z>x\}$
3. $P\left\{\right|Z|<x\}=2P\{Z<x\}-1$
   

The mode of a continuous random variable having density $f$ is the value of $x$for which $f\left(x\right)$ attains its maximum. Compute the mode of $x$in cases $\left(a\right),\left(b\right)$ and $\left(c\right)$of Theoretical Exercise$5.13$ 

Compute the hazard rate function of $X$ when $X$is uniformly distributed over.$(0, a)$

Find the probability density function of Y = eX when X is normally distributed with parameters μ and σ2. The random variable Y is said to have a lognormal distribution (since log Y has a normal distribution) with parameters μ and σ2. 

Let X have probability density f X. Find the probability density function of the random variable Y defined by Y = a X + b. 

Let X be a continuous random variable having cumulative distribution function F. Define the random variable Y by Y = F(X). Show that Y is uniformly distributed over (0, 1). 

 Let X and Y be independent random variables that are both equally likely to be either 1, 2, . . . ,(10)N, where N is very large. Let D denote the greatest common divisor of X and Y, and let Q k = P{D = k}. 

(a) Give a heuristic argument that Q k = 1 k2 Q1. Hint: Note that in order for D to equal k, k must divide both X and Y and also X/k, and Y/k must be relatively prime. (That is, X/k, and Y/k must have a greatest common divisor equal to 1.) (b) Use part (a) to show that Q1 = P{X and Y are relatively prime} = 1 q k=1 1/k2 It is a well-known identity that !q 1 1/k2 = π2/6, so Q1 = 6/π2. (In number theory, this is known as the Legendre theorem.) (c) Now argue that Q1 = "q i=1  P2 i − 1 P2 i  where Pi is the smallest prime greater than 1. Hint: X and Y will be relatively prime if they have no common prime factors. Hence, from part (b), we see that Problem 11 of Chapter 4 is that X and Y are relatively prime if XY has no multiple prime factors.) 

If has a hazard rate function${\lambda }_X\left(t\right)$, compute the hazard rate function of $a X$where $a$is a positive constant.

Verify that the gamma density function integrates to$1$.

 If $X$ is an exponential random variable with a mean$1/\lambda$, show that

$E\left[X^k\right]=\frac{k!}{{\lambda }^k} k=1,2,\dots$

Hint: Make use of the gamma density function to evaluate the preceding.

Show that $\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$

Hint: $\Gamma \left(\frac{1}{2}\right)={\int }_0^{\infty }{e}^{-x}{x}^{-1/2}dx$ Make the change of variables $y=\sqrt{2x}$ and then relate the resulting expression to the normal distribution.

Verify that$Var\left(X\right)=\frac{\alpha }{{\lambda }^2}$ when$X$ is a gamma random variable with parameters $\alpha$and $\lambda$

Compute the hazard rate function of a gamma random variable with parameters $(\alpha ,\lambda )$and show it is increasing when $\alpha$$\ge 1$ and decreasing when $\alpha \le 1$

Show that a plot of $\log \left(\log (1-\mathrm{F}(\mathrm{x}){)}^{-1}\right)$ against $\log \left(\mathrm{X}\right)$ will be a straight line with slope $\mathrm{\beta }$ when $\mathrm{F}(-)$ is a Weibull distribution function. Show also that approximately $63.2$ percent of all observations from such a distribution will be less than $\mathrm{\alpha }$. Assume that $\mathrm{v}=0$.

If $\mathrm{x}$ is a beta random variable with parameters $\mathrm{a}$ and $\mathrm{b}$, show that

$\mathrm{E}\left[\mathrm{X}\right]=\frac{\mathrm{a}}{\mathrm{a}+\mathrm{b}}$

$Var\left(\mathrm{X}\right)=\frac{\mathrm{ab}}{(\mathrm{a}+\mathrm{b}{)}^2(\mathrm{a}+\mathrm{b}+1)}$

If $\mathrm{x}$ is uniformly distributed over $(\mathrm{a},\mathrm{b})$, what random variable, having a linear relation with $\mathrm{x}$, is uniformly distributed over $(0,1)$$?$

 Consider the beta distribution with parameters $(\mathrm{a}, \mathrm{b})$. Show that

(a) when $\mathrm{a}>1$ and $\mathrm{b}>1$, the density is unimodal (that is, it has a unique mode) with mode equal to $(\mathrm{a}-1) /(\mathrm{a}+\mathrm{b}-2)$

(b) when $\mathrm{a} \le 1$, $\mathrm{b}\le 1$, and $\mathrm{a}+\mathrm{b}<2$, the density is either unimodal with mode at $0$ or $1$or U-shaped with modes at both$0$ and$1$ ;

(c) when $\mathrm{a}=1=\mathrm{b}$, all points in $[0,1]$ are modes.

Compute the hazard rate function of a Weibull random variable and show it is increasing when $𝛽\ge 1$ and decreasing when $𝛽\le 1$

Let $Y=\frac{X-𝜈}{𝛼}$.

Show that if X is a Weibull random variable with parameters ν, α, and β, then Y is an exponential random variable with parameter λ = 1 and vice versa.

If $X$ is an exponential random variable with mean $1/\lambda$, show that

$E\left[X^k\right]=\frac{k!}{{\lambda }^k} k=1,2,\dots$

Hint: Make use of the gamma density function to evaluate the preceding.

If $\mathit{X}$ is an exponential random variable with a mean $\frac{\mathbf{1}}{\mathbf{\lambda }}$, show that

$\mathit{E}\mathbf{\left[}{\mathbf{X}}^{\mathbf{k}}\mathbf{\right]}\mathbf{=}\frac{\mathbf{k}\mathbf{!}}{{\mathbf{\lambda }}^{\mathbf{k}}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathit{k}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{\dots }$

 For some constant c, the random variable X has the probability density function f(x) =  c x n 0 < x < 1 0 otherwise Find (a) c and 

(b) P{X > x}, 0 < x < 1. 

For some constant c, the random variable X has the probability density function:

$f\left(x\right)=\left\{\begin{array}{l}c{x}^4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0<x<2\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ otherwise }\end{array}\right.$

Find

1. $E\left[X\right]$ and 
2. $Var\left(X\right)$
   

Your company must make a sealed bid for a construction project. If you succeed in winning the contract (by having the lowest bid), then you plan to pay another firm $100,000 to do the work. If you believe that the minimum bid (in thousands of dollars) of the other participating companies can be modeled as the value of a random variable that is uniformly distributed on (70, 140), how much should you bid to maximize your expected profit? 

The random variable X is said to be a discrete uniform random variable on the integers 1, 2, . . . , n if P{X = i } = 1 n i = 1, 2, . .  , n For any nonnegative real number x, let In t(x) (sometimes written as [x]) be the largest integer that is less than or equal to x. Show that if U is a uniform random variable on (0, 1), then X = In t  (n U) + 1 is a discrete uniform random variable on 1, . . .  , n. 

Prove Theorem 7.1 when g(x) is a decreasing function. 

To be a winner in a certain game, you must be successful in three successive rounds. The game depends on the value of U, a uniform random variable on $(0,1)$. If $U>.1$, then you are successful in round $1$; if $U>1$, then you are successful in round $2$; and if $U>3$, then you are successful in round $3$.

 (a) Find the probability that you are successful in round $1$.

 (b) Find the conditional probability that you are successful in round $2$ given that you were successful in round $1$.

 (c) Find the conditional probability that you are successful in round $3$ given that you were successful in rounds $1 and 2$ 

 (d) Find the probability that you are a winner 

A randomly chosen $IQ$ test taker obtains a score that is approximately a normal random variable with mean $100$ and standard deviation $15$. What is the probability that the score of such a person is

(a) more than 125; 

(b) between $90$ and $110$?

Suppose that the travel time from your home to your office is normally distributed with mean $40$ minutes and standard deviation $7$ minutes. If you want to be $95$ percent certain that you will not be late for an office appointment at $1$ p.m., what is the latest time that you should leave home?

The life of a certain type of automobile tire is normally distributed with mean $34,000$miles and standard deviation $4,000$miles.

(a) What is the probability that such a tire lasts more than $40,000$miles?

(b) What is the probability that it lasts between $30,000$and$35,000$ miles?

(c) Given that it has survived $30,000$miles, what is the conditional probability that the tire survives another $10,000$miles?

The number of minutes of playing time of a certain high school basketball player in a randomly chosen game is a random variable whose probability density function is given in the following figure: 

Find the probability that the player plays 

(a) more than $15$ minutes;

(b) between $20 and 35$ minutes;

(c) less than $30$ minutes;

(d) more than $36$ minutes 

Your company must make a sealed bid for a construction project. If you succeed in winning the contract (by having the lowest bid), then you plan to pay another firm $\$100,000$ to do the work. If you believe that the minimum bid (in thousands of dollars) of the other participating companies can be modeled as the value of a random variable that is uniformly distributed on $(70, 140)$, how much should you bid to maximize your expected profit? 

 A standard Cauchy random variable has density function 

$f\left(x\right)=\frac{1}{\pi \left(1+x^2\right)}-\mathrm{\infty }<x<\mathrm{\infty }$

Show that if X is a standard Cauchy random variable, then 1/X is also a standard Cauchy random variable. 

A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, then you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the probability that 

(a) you are winning after 34 bets; 

(b) you are winning after 1000 bets; 

(c) you are winning after 100,000 bets 

Assume that each roll of the roulette ball is equally likely to land on any of the 38 numbers 

There are two types of batteries in a bin. When in use, type i batteries last (in hours) an exponentially distributed time with rate ${\lambda }_i,i=1,2$. A battery that is randomly chosen from the bin will be a type i battery with probability pi, $p_i,\sum _{i=1}^2 p_i=1$. If a randomly chosen battery is still operating after t hours of use, what is the probability that it will still be operating after an additional s hours? 

Evidence concerning the guilt or innocence of a defendant in a criminal investigation can be summarized by the value of an exponential random variable X whose mean μ depends on whether the defendant is guilty. If innocent, μ = 1; if guilty, μ = 2. The deciding judge will rule the defendant guilty if X > c for some suitably chosen value of c. 

(a) If the judge wants to be 95 percent certain that an innocent man will not be convicted, what should be the value of c? 

(b) Using the value of c found in part (a), what is the probability that a guilty defendant will be convicted? 

The annual rainfall in Cleveland, Ohio, is approximately a normal random variable with mean 40.2 inches and standard deviation 8.4 inches. What is the probability that (a) next year’s rainfall will exceed 44 inches? (b) the yearly rainfalls in exactly 3 of the next 7 years will exceed 44 inches? Assume that if Ai is the event that the rainfall exceeds 44 inches in year i (from now), then the events Ai, i Ú 1, are independent. 

The following table uses $1992$ data concerning the percentages of male and female full-time workers whose

annual salaries fall into different ranges:

Suppose that random samples of 200 male and 200 female full-time workers are chosen. Approximate the probability

that

(a) at least $70$ of the women earn $\backslash (25,000$ or more;

(b) at most $60$ percent of the men earn $\backslash )25,000$ or more;

(c) at least three-fourths of the men and at least half the women earn $\$20,000$ or more.

At a certain bank, the amount of time that a customer spends being served by a teller is an exponential random variable with mean $5$ minutes. If there is a customer in service when you enter the bank, what is the probability that he or she will still be with the teller after an additional $4$ minutes?

Suppose that the cumulative distribution function of the random variable $X$ is given by

$F\left(x\right)=1-e^{-x^2}x>0$

Evaluate $\text{ (a) }P[X>2];\text{ (b) }P[1<X<3)$; (c) the hazard rate function of $F_i\text{ (d) }E\left[X\right]\text{; (e) }\mathrm{Var}\left(X\right)$.

Hint: For parts $\left(d\right)$ and $\left(e\right)$, you might want to make use of the results of Theoretical Exercise  $5.5$.