Exam 8: Bivariate Distributions

Suppose that five numbers are chosen from 1 to 60 (inclusive, without replacement). If X is the number of even numbers, and Y is the number of powers of 3 in the list of the five chosen, find the joint probability mass function of X and Y.

Free

(Short Answer)

4.9/5

(29)

Question 1

Correct Answer:

Verified

$f ( x , y ) = \left\{ \begin{array} { l l } \frac { \left( \begin{array} { c } 30 \\x\end{array} \right) \left( \begin{array} { l } 4 \\y\end{array} \right) \left( \begin{array} { c } 26 \\5 - ( x + y )\end{array} \right) } { \left( \begin{array} { c } 60 \\5\end{array} \right) } & x = 1 , \ldots , 5 ; y = 1 , \ldots , 4 , x + y \leq 5 \\0 & \text { elsewhere }\end{array} \right.$

Let X and Y be uniform random variables over (0,2) and (0,4), respectively. Find the probability that |X-Y|<2.

Free

(Short Answer)

4.7/5

(34)

Question 2

Correct Answer:

Verified

.75.

LLet X and Y be independent random variables uniformly distributed over (0,1). Find the distribution function and probability density function of U=XY.

Free

(Essay)

4.8/5

(35)

Question 3

Correct Answer:

Verified

The distribution function is $F _ { U } ( u ) = \left\{ \begin{array} { l l } o & u \leq 0 \\u - u \ln ( u ) & 0 < u < 1 \\1 & u \geq 1\end{array} \right.$ The probability density function is $f ( u ) = \left\{ \begin{array} { l l } - \ln ( u ) & 0 < u < 1 \\0 & \text { else }\end{array} \right.$

To invite to a dinner, the White House randomly selects 9 athletes from a group of 5 runners, 7 gymnasts, and 6 boxers. Let X be the number of gymnasts invited, and Y the number of boxers invited. Find the joint probability mass function of X and Y and the conditional probability density function of X given Y=2.

(Essay)

4.9/5

(34)

Question 4

Let X and Y have joint probability density function $f ( x , y ) = \left\{ \begin{array} { l l } \frac { 4 } { 11 } \left( 2 x + y ^ { 2 } \right) & 1 < x < 2 , x < y < 2 \\0 & \text { elsewhere }\end{array} \right.$ Find the marginal probability density functions of X and Y and P(0<(Y-X)/(1-x)<2).

(Essay)

4.8/5

(32)

Question 5

A game is played by throwing a bean bag onto a circular game board of radius 3m. There is a region, a .5m-by-.5m square, in the center of the board. If the bag lands at a random location on the board, find the probability that it lands on the square.

(Short Answer)

4.8/5

(25)

Question 6

A certain aircraft can only fly if both of its two engines are functioning properly. This plane is a home project and so has two different engines. The lifetimes of the engines, in years, are given by random variable X and Y and the joint probability density function of X and Y is given by $f ( x , y ) = \left\{ \begin{array} { l l } \frac { x ^ { 2 } + y ^ { 2 } } { 26 } & 0 < X < 2 \text { and } 0 < Y < 3 \\0 & \text { elsewhere. }\end{array} \right.$ (a) Find the probability that the airplane is capable of flying for more than 1 year. (b) Find the marginal distribution function of Y.

(Essay)

4.9/5

(27)

Question 7

Two numbers are picked in succession from 1 to 4 (inclusive, with replacement). Let X denote the minimum of the numbers and Y the result of the first number minus the second. (a) Give the joint probability mass function of X and Y. (b) Give the marginal probability mass function of X, and (c) find E(Y+2).

(Essay)

4.9/5

(39)

Question 8

Let X and Y be two random variables with joint probability density function $f ( x , y ) = \left\{ \begin{array} { l l } A x ^ { 2 } y & 0 < x < 1,0 < y < 2 \\0 & \text { elsewhere }\end{array} \right.$ (a) Find A. (b) Find the marginal distribution function of X.

(Essay)

4.8/5

(28)

Question 9

A point is selected at random in the square [-1,1]×[-1,1]. Let X and Y represent the x- and y-coordinates of the point, respectively. What is probability of the following events, (a) $E = \left\{ x + y < \frac { 1 } { 2 } \right\}$ ; (b) F={|X|+|y|<1}?

(Short Answer)

4.8/5

(39)

Question 10

Suppose that weights of batteries of brand A are normally distributed with mean 35 grams and standard deviation 3 grams. Suppose that weights of brand B batteries are normally distributed with mean 38 grams and standard deviation 4 grams. Find the probability that a randomly selected battery of brand A is heavier than a randomly selected battery of brand B.

(Short Answer)

4.9/5

(32)

Question 11

A die is rolled successively. Let X represent the number of 1s in the first 8 rolls and Y the number of 1s in the next 12 rolls. Find the joint probability mass function of X and Y.

(Essay)

4.9/5

(36)

Question 12

A point is selected at random from the trapezoid bounded by the lines y=0, y=x+2, y=-x+2, and y=1. Let X be the x-coordinate and Y the y-coordinate of the point. Find P(X>1|Y>.5). Are X and Y independent?

(Essay)

4.8/5

(43)

Question 13

In an experiment a standard die is rolled twice. Let X be the sum of the rolls and Y be the first roll minus the second. Show that E(XY)=E(X)E(Y), but X and Y are not independent.

(Essay)

4.8/5

(38)

Question 14

Suppose that five numbers are chosen from 1 to 60 (inclusive, without replacement). If X is the number of even numbers, and Y is the number of powers of 3 in the list of the five chosen, find the joint probability mass function of X and Y.

Let X and Y be uniform random variables over (0,2) and (0,4), respectively. Find the probability that |X-Y|<2.

LLet X and Y be independent random variables uniformly distributed over (0,1). Find the distribution function and probability density function of U=XY.

Let X and Y have joint probability density function $f ( x , y ) = \left\{ \begin{array} { l l } \frac { 4 } { 11 } \left( 2 x + y ^ { 2 } \right) & 1 < x < 2 , x < y < 2 \\0 & \text { elsewhere }\end{array} \right.$ Find the marginal probability density functions of X and Y and P(0<(Y-X)/(1-x)<2).

A game is played by throwing a bean bag onto a circular game board of radius 3m. There is a region, a .5m-by-.5m square, in the center of the board. If the bag lands at a random location on the board, find the probability that it lands on the square.

Let X and Y be two random variables with joint probability density function $f ( x , y ) = \left\{ \begin{array} { l l } A x ^ { 2 } y & 0 < x < 1,0 < y < 2 \\0 & \text { elsewhere }\end{array} \right.$ (a) Find A. (b) Find the marginal distribution function of X.

A point is selected at random in the square [-1,1]×[-1,1]. Let X and Y represent the x- and y-coordinates of the point, respectively. What is probability of the following events, (a) $E = \left\{ x + y < \frac { 1 } { 2 } \right\}$ ; (b) F={|X|+|y|<1}?

A die is rolled successively. Let X represent the number of 1s in the first 8 rolls and Y the number of 1s in the next 12 rolls. Find the joint probability mass function of X and Y.

A point is selected at random from the trapezoid bounded by the lines y=0, y=x+2, y=-x+2, and y=1. Let X be the x-coordinate and Y the y-coordinate of the point. Find P(X>1|Y>.5). Are X and Y independent?

In an experiment a standard die is rolled twice. Let X be the sum of the rolls and Y be the first roll minus the second. Show that E(XY)=E(X)E(Y), but X and Y are not independent.

Axioms of Peobability

Combinatorial Metods

Conditional Probability and Independence

Distribution Functions and Discrete Random Variables

Apecial Discrete Distributions

Continuous Random Variables

Special Continuous Distribution

Multivariate Distributions

More Expectations and Variances

Sums of Independent Random Variables and Limit Theorems

Stocastic Processes

Filters

Exam 8: Bivariate Distributions

Suppose that five numbers are chosen from 1 to 60 (inclusive, without replacement). If X is the number of even numbers, and Y is the number of powers of 3 in the list of the five chosen, find the joint probability mass function of X and Y.

Let X and Y be uniform random variables over (0,2) and (0,4), respectively. Find the probability that |X-Y|<2.

LLet X and Y be independent random variables uniformly distributed over (0,1). Find the distribution function and probability density function of U=XY.

A game is played by throwing a bean bag onto a circular game board of radius 3m. There is a region, a .5m-by-.5m square, in the center of the board. If the bag lands at a random location on the board, find the probability that it lands on the square.

A point is selected at random in the square [-1,1]×[-1,1]. Let X and Y represent the x- and y-coordinates of the point, respectively. What is probability of the following events, (a) E={x+y<12}E = \left\{ x + y < \frac { 1 } { 2 } \right\}E={x+y<21​} ; (b) F={|X|+|y|<1}?

A die is rolled successively. Let X represent the number of 1s in the first 8 rolls and Y the number of 1s in the next 12 rolls. Find the joint probability mass function of X and Y.

A point is selected at random from the trapezoid bounded by the lines y=0, y=x+2, y=-x+2, and y=1. Let X be the x-coordinate and Y the y-coordinate of the point. Find P(X>1|Y>.5). Are X and Y independent?

In an experiment a standard die is rolled twice. Let X be the sum of the rolls and Y be the first roll minus the second. Show that E(XY)=E(X)E(Y), but X and Y are not independent.

Axioms of Peobability

Combinatorial Metods

Conditional Probability and Independence

Distribution Functions and Discrete Random Variables

Apecial Discrete Distributions

Continuous Random Variables

Special Continuous Distribution

Multivariate Distributions

More Expectations and Variances

Sums of Independent Random Variables and Limit Theorems

Stocastic Processes

Filters

A point is selected at random in the square [-1,1]×[-1,1]. Let X and Y represent the x- and y-coordinates of the point, respectively. What is probability of the following events, (a) $E = \left\{ x + y < \frac { 1 } { 2 } \right\}$ ; (b) F={|X|+|y|<1}?