Jointly Distributed Random Variables

The time that it takes to service a car is an exponential random variable with rate $1$.

(a) If A. J. brings his car in at time $0$ and M. J. brings her car in at time t, what is the probability that M. J.’s car is ready before A. J.’s car? (Assume that service times are independent and service begins upon arrival of the car.)

(b) If both cars are brought in at time 0, with work starting on M. J.’s car only when A. J.’s car has been completely serviced, what is the probability that M. J.’s car is ready before time $2$? 

Repeat Problem $6.3a$ when the ball selected is replaced in the urn before the next selection 

Consider a sequence of independent Bernoulli trials, each of which is a success with probability p. Let X1 be the number of failures preceding the first success, and let X2 be the number of failures between the first two successes. Find the joint mass function of X1 and X2.

The number of people who enter a drugstore in a given hour is a Poisson random variable with parameter λ = 10. Compute the conditional probability that at most 3 men entered the drugstore, given that 10 women entered in that hour. What assumptions have you made? 

A man and a woman agree to meet at a certain location about 12:30 p.m. If the man arrives at a time uniformly distributed between 12:15 and 12:45, and if the woman independently arrives at a time uniformly distributed between 12:00 and 1 p.m., find the probability that the first to arrive waits no longer than 5 minutes. What is the probability that the man arrives first? 

A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N1 the number of tests made until the first defective is identified and by N2 the number of additional tests until the second defective is identified. Find the joint probability mass function of N1 and N2. 

$$\begin{gathered} The joint probability density function of X and Y is given by \\ f(x,y)=e-(x+y)0\dots x<q,0\dots y<q \\ Find \\ \left(a\right)PX<Yand \\ \left(b\right)PX<a \end{gathered}$$

Repeat Problem $6.2$ when the ball selected is replaced in the urn before the next selection.

In Problem $6.2$, suppose that the white balls are numbered, and let ${\mathrm{Y}}_{\mathrm{i}}$equal $1$ if the $\mathrm{i}$th white ball is selected and$0$ otherwise. Find the joint probability mass function of

(a)${\mathrm{Y}}_1,{\mathrm{Y}}_2$

(b) $\mathrm{Y\_}\left\{1\right\}, \mathrm{Y\_}\left\{2\right\}, \mathrm{Y\_}\left\{3\right\}$

Suppose that $3$ balls are chosen without replacement from an urn consisting of $5$white and$8$ red balls. Let ${\mathrm{X}}_i$ equal $1$ if the $\mathrm{i}$th ball selected is white, and let it equal $0$otherwise. Give the joint probability mass function of

(a) ${\mathrm{X}}_1,{\mathrm{X}}_2$;

(b)${\mathrm{X}}_1,{\mathrm{X}}_2,{\mathrm{X}}_3$ .

Two fair dice are rolled. Find the joint probability mass function of $\mathrm{X}$and $\mathrm{Y}$ when

(a) $\mathrm{X}$is the largest value obtained on any die and$\mathrm{Y}$ is the sum of the values;

(b) $\mathrm{X}$ is the value on the first die and $\mathrm{Y}$ is the larger of the two values;

(c) $\mathrm{X}$ is the smallest and $\mathrm{Y}$ is the largest value obtained on the dice.

An ambulance travels back and forth at a constant speed along a road of length L. At a certain moment of time, an accident occurs at a point uniformly distributed on the road. [That is, the distance of the point from one of the fixed ends of the road is uniformly distributed over (0, L).] Assuming that the ambulance’s location at the moment of the accident is also uniformly distributed, and assuming independence of the variables, compute the distribution of the distance of the ambulance from the accident.

Suppose that${10}^6$people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over$\left(0,{{10}^6}_{}\right)$. Let N denote the number that arrive in the first hour. Find an approximation for$P\left\{N=i\right\}$.

Three points X₁, X₂, X₃ are selected at random on a line L. What is the probability that X₂ lies between X₁ and X₃? 

The joint density function of $X$ and $Y$ is

(a) Are $X$ and $Y$ independent?

(b) Find the density function of $X$.

(c) Find$P\{X+Y<1\}$.

The random variables $X and Y$ have joint density function $f(x, y) = 12xy(1 - x) 0 < x < 1, 0 < y < 1$ and equal to$0$ otherwise.

(a) Are $X and Y$ independent?

(b) Find $E\left[X\right].$ 

(c) Find $E\left[Y\right].$

(d) Find $Var\left(X\right)$. 

(e) Find $Var\left(Y\right).$ 

The joint density function of X and Y is $f(x, y) = \left\{\begin{array}{l}x+y 0 < x < 1, 0 < y < 1\\ 0 0 otherwise\end{array}\right.$

(a) Are X and Y independent?

(b) Find the density function of X.

(c) Find $P[X+Y<1].$

Let $f(x, y) = 24xy 0 \le x \le 1, 0 \le y \le 1, 0 \le x + y \le 1$ and let it equal 0 otherwise.

(a) Show that $f(x, y)$ is a joint probability density function.

(b) Find $E\left[X\right]$. 

(c) Find $E\left[Y\right]$. 

Suppose that n points are independently chosen at random on the circumference of a circle, and we want the probability that they all lie in some semicircle. That is, we want the probability that there is a line passing through the center of the circle such that all the points are on one side of that line, as shown in the following diagram: 

Let P1, ... ,Pn denote the n points. Let A denote the event that all the points are contained in some semicircle, and let Ai be the event that all the points lie in the semicircle beginning at the point Pi and going clockwise for 180◦, i = 1, ... , n.

 (a) Express A in terms of the Ai.

 (b) Are the Ai mutually exclusive?

 (c) Find P(A). 

Two points are selected randomly on a line of length L so as to be on opposite sides of the midpoint of the line. [In other words, the two points X and Y are independent random variables such that X is uniformly distributed over (0, L/2) and Y is uniformly distributed over (L/2, L).] Find the probability that the distance between the two points is greater than L/3 

Show that f(x, y) = 1/x, 0 < y < x < 1, is a joint density function. Assuming that f is the joint density function of X, Y, find

 (a) the marginal density of Y; 

(b) the marginal density of X;

 (c) E[X]; (d) E[Y].  

Consider independent trials, each of which results in outcome i, i = $0,1, ....,k$, with probability $p_i,\sum _{i=0}^K{p}_i=1$ . Let N denote the number of trials needed to obtain an outcome that is not equal to $0$, and let X be that outcome.

(a) Find $P\{N=n\},n\ge 1$ 

(b) Find  $P\{X=j\},j=1,\dots ,k$

(c) Show that $P\{N=n,X=j\}=P\{N=n\}P\{X=j\}$.

(d) Is it intuitive to you that N is independent of X? 

(e) Is it intuitive to you that X is independent of N? 

Jill’s bowling scores are approximately normally distributed with mean $170$ and standard deviation $20$, while Jack’s scores are approximately normally distributed with mean $160$ and standard deviation $15$. If Jack and Jill each bowl one game, then assuming that their scores are independent random variables, approximate the probability that

(a) Jack’s score is higher; 

(b) the total of their scores is above $350$ 

The joint density function of $\mathbf{X}$ and $\mathbf{Y}$ is

$\mathit{f}\left(x,y\right)\mathbf{=}\{\begin{array}{l}x+y 0<x<1,0<y<1\\ 0 \mathrm{otherwise}\end{array}$

1. Are $\mathbf{X}$ and $\mathbf{Y}$ independent?
2. Find the density function of $\mathbf{X}$.
3. Find $\mathit{P}\left\{X+Y<1\right\}$.

If $X1/X2$ are independent exponential random variables with respective parameters $\lambda 1$and $\lambda 2$, find the distribution of $Z=X1/X2$. Also compute $P\left\{X1<X2\right\}$. 

The gross weekly sales at a certain restaurant are a normal random variable with mean$\backslash (2200$ and standard deviation $\backslash )230$ . What is the probability that

(a) the total gross sales over the next $2$ weeks exceeds $\backslash (5000$;

(b) weekly sales exceed $\backslash )2000$ in at least $2$ of the next $3$ weeks? What independence assumptions have you made?

Monthly sales are independent normal random variables with mean $100$ and standard deviation $5$.

(a) Find the probability that exactly $3$ of the next $6$ months have sales greater than $100$.

(b) Find the probability that the total of the sales in the next $4$ months is greater than $420$. 

The expected number of typographical errors on a page of a certain magazine is $.2$. What is the probability that an article of $10$ pages contains (a) $0$ and (b) $2$ or more typographical errors? Explain your reasoning!

The monthly worldwide average number of airplane crashes of commercial airlines is $2.2$. What is the probability that there will be (a) more than $2$ such accidents in the next month? (b) more than $4$ such accidents in the next $2$ months? (c) more than $5$ such accidents in the next $3$ months? Explain your reasoning!

In Problem $6.4$, calculate the conditional probability mass function of $X_1$ given that

(a) $X_2=1;$ 

(b) $X_1=0$ 

In Problem $6.3$, calculate the conditional probability mass function of  $Y_1$ given that

(a) $Y_2=1;$ 

(b) $Y_2=0.$ 

Choose a number X at random from the set of numbers $\left\{1,2,3,4,5\right\}$. Now choose a number at random from the subset no larger than X, that is, from $\left\{1 ... ,X\right\}$. Call this second number Y.

(a) Find the joint mass function of X and Y.

(b) Find the conditional mass function of X given that Y = i. Do it for i = $1,2,3,4,5$.

(c) Are X and Y independent? Why? 

Two dice are rolled. Let X and Y denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of Y given X = i, for i = $1,2, ... ,6$. Are X and Y independent? Why? 

The joint probability mass function of X and Y is given by 

$$\begin{gathered} p(1,1)=\frac{1}{8} p(1,2)=\frac{1}{4} \\ p(2,1)=\frac{1}{8} p(2,2)=\frac{1}{2} \end{gathered}$$

In Problem $6.5$, calculate the conditional probability mass function of Y1 given that

(a) $Y_2=1$

(b) $Y_2=0$

If $X_1,X_2,X_3$ are independent random variables that are uniformly distributed over $\left(0,1\right)$, compute the probability that the largest of the three is greater than the sum of the other two. 

A complex machine is able to operate effectively as long as at least $3$ of its $5$ motors are functioning. If each motor independently functions for a random amount of time with density function$f\left(x\right) = x{e}^{-x}, x>0,$ compute the density function of the length of time that the machine functions. 

If $3$ trucks break down at points randomly distributed on a road of length L, find the probability that no $2$ of the trucks are within a distance d of each other when  $d\le L/2$.

Consider a sample of size $5$ from a uniform distribution over $(0,1)$. Compute the probability that the median is in the interval $\left(\frac{1}{4},\frac{3}{4}\right)$. 

If $X_1,X_2,X_3,X_4,X_5$ are independent and identically distributed exponential random variables with the parameter $\lambda$, compute

(a) $P\{min(X_1,...,X_5)\le a\}$;

(b) $P\{max(X_1,...,X_5)\le a.$ 

Let $X_{\left(1\right)}, X_{\left(2\right)}, ... , X_{\left(n\right)}$ be the order statistics of a set of n independent uniform $(0, 1)$ random variables. Find the conditional distribution of $X_{\left(n\right)}$ given that $X_{\left(1\right)=S_1},X_{\left(2\right)=S_2},....,X_{(n-1)=S(n-1)}$.

Let $Z_1$ and $Z_2$ be independent standard normal random variables. Show that X, Y has a bivariate normal distribution when $X =Z_1, Y = Z_1 + Z_2$.

Derive the distribution of the range of a sample of size $2$ from a distribution having density function $f\left(x\right) = 2x, 0 < x < 1.$ 

Let X and Y denote the coordinates of a point uniformly chosen in the circle of radius $1$ centered at the origin. That is, their joint density is $f(x, y) = \frac{1}{\mathrm{\pi }}{x}^2+y^2\le 1$.

Find the joint density function of the polar coordinates $R = {(X^2 + Y^2)}^{1/2}$ and $\theta ={\tan }^{-1}Y/X$. 

If X and Y are independent random variables both uniformly distributed over $(0,1)$, find the joint density function of $R=\sqrt{X^2+Y^2}, \theta ={\tan }^{-1} Y/X$. 

If U is uniform on $(0,2\mathrm{\pi })$ and Z, independent of U, is exponential with rate $1$, show directly (without using the results of Example $7$b) that X and Y defined by

$$\begin{gathered} X=\sqrt{2Z}\cos U \\ Y=\sqrt{2Z}\sin U \end{gathered}$$

 are independent standard normal random variables. 

X and Y have joint density function

$f(x,y)=\frac{1}{x^2{y}^2} x\ge 1,y\ge 1$

(a) Compute the joint density function of U = XY, V = X/Y. 

(b) What are the marginal densities? 

If X and Y are independent and identically distributed uniform random variables on$(0,1)$, compute the joint density of

$$\begin{gathered} \left(a\right) U = X + Y, V = X/Y; \\ \left(b\right) U = X, V = X/Y; \\ \left(c\right) U = X + Y, V = X/(X + Y). \end{gathered}$$

Repeat Problem $6.56$ when X and Y are independent exponential random variables, each with parameter $\lambda = 1$. 

If X1 and X2 are independent exponential random variables, each having parameter $\lambda$, find the joint density function of $Y_1 = X_1 + X_2$ and $Y_2=e^{X_1}$ .