Practice set 5.2

For all, sketch the regions of integration!!!

1

Suppose \(X\) and \(Y\) are jointly continuous with joint pdf given by:

\[ f(x, y) = \begin{cases} 1, & 0 \le x \le 1, \; 0 \le y \le 1, \\ 0, & \text{elsewhere.} \end{cases} \]

A

What is \(P(X - Y > 0.5)\)?

B

What is \(P(XY < 0.5)\)?

2

Let X and Y have joint density function

\[f(x, y) = \begin{cases} e^{-(x + y)}, & x > 0, \; y > 0, \\ 0, & \text{elsewhere.} \end{cases}\]

A

Find the joint CDF.

B

What is \(P(X < 1, \, Y > 5)\)?

C

What is \(P(X + Y < 3)\)?

3

Let X and Y have the joint probability density function given by

\[f(x, y) = \begin{cases} kxy, & 0 \le x \le 1, \; 0 \le y \le 1, \\ 0, & \text{elsewhere.} \end{cases}\]

A

Find the value of \(k\) that makes this a probability density function.

B

Find the joint cumulative distribution function for X and Y.

C

Find \(P(X \le 1/2, \, Y \le 3/4)\).

D

Find \(P(X \le 3/4, Y < 1.5)\).

4

Let X and Y have the joint probability density function given by

\[ f(x, y) = \begin{cases} k(1 - y), & 0 \le x \le 1, \; 0 \le y \le 1, \\ 0, & \text{elsewhere.} \end{cases} \]

A

Find the value of \(k\) that makes this a probability density function.

B

Find \(P(X<Y)\).

5

Let X and Y denote the proportions of two different types of components in a sample from a mixture of chemicals used as an insecticide. Suppose that X and Y have the joint density function given by

\[ f(x, y) = \begin{cases} 2, & 0 \le x \le 1,\; 0 \le y \le 1,\; 0 \le x + y \le 1, \\ 0, & \text{elsewhere.} \end{cases} \]

(Notice that \(X + Y \le 1\) because the random variables denote proportions within the same sample.)

A

Find the joint cumulative distribution function.

B

Find \(P(X \le 3/4,\, Y \le 3/4).\)

C

Find \(P(X \le 1/2,\, Y \le 1/4).\)

D

Find \(P(X\leq 3/4, Y \leq 2).\)

6

Suppose that the random variables X and Y have joint probability density function given by

\[ f(x, y) = \begin{cases} 6x^2y, & 0 \le x \le y,\; x + y \le 2, \\ 0, & \text{elsewhere.} \end{cases} \]

A

Verify that this is a valid joint density function.

B

What is the probability that \(X + Y\) is less than 1?

7

The management at a fast-food outlet is interested in the joint behavior of the random variables \(X\) and \(Y\), defined as the total time between a customer’s arrival at the drive-thru and departure from the service window, and the time a customer waits in line before reaching the service window, respectively.

Because \(Y\) includes the time a customer waits in line, we must have \(Y \le X\).

The relative frequency distribution of observed values of \(X\) and \(Y\) can be modeled by the probability density function

\[ f(x, y) = \begin{cases} e^{-x}, & 0 \le y \le x < \infty, \\ 0, & \text{elsewhere.} \end{cases} \]

with time measured in minutes.

A

Find the joint CDF.

B

Use the joint CDF to find \(P(X < 2,\, Y > 1)\)

C

Use the joint CDF to find \(P(X>2, Y < 1)\).

D

Find \(P(X>2, Y>1)\).

E

Find \(P(X \ge 2Y)\)

8

Let \((X, Y)\) denote the coordinates of a point chosen at random inside a unit circle whose center is at the origin.

That is, \(X\) and \(Y\) have a joint density function given by

\[ f(x, y) = \begin{cases} \dfrac{1}{\pi}, & x^2 + y^2 \le 1, \\ 0, & \text{elsewhere.} \end{cases} \]

Find \(P(X \le Y).\)

9

Suppose \(X\) and \(Y\) are uniformly distributed over the triangular region shown below:

A

What is the joint probability density function?

B

Find \(P(Y>|X|)\).