Deck 8: Bivariate Distributions

Question 1

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.

Answer 1

(a) .513
(b) $F _ { Y } ( y ) = \left\{ \begin{array} { l l } 0 & y < 0 \\\frac { 24 y + 2 y ^ { 2 } } { 78 } & 0 < y < 3 \\1 & \mathbf { y } > 3\end{array} \right.$

Question 2

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

Answer 2

.75.

Question 3

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.

Answer 3

(a) $p ( x , y ) = \left\{ \begin{array} { l l } \frac { \left( \begin{array} { l } 7 \\x\end{array} \right) \left( \begin{array} { l } 6 \\y\end{array} \right) \left( \begin{array} { c } 5 \\0 - ( x + y )\end{array} \right) } { \left( \begin{array} { c } 18 \\9\end{array} \right) } & x = 0,1 , \ldots , 7 , y = 0,1 , \text { dots } , 6,4 \leq x + y \leq 9 \\0 & \text { otherwise. }\end{array} \right.$
(b) $p _ { X \mid Y } ( x \mid 2 ) = \left\{ \begin{array} { l l } \frac { \left( \begin{array} { l } 7 \\x\end{array} \right) \left( \begin{array} { l } 6 \\2\end{array} \right) \left( \begin{array} { c } 5 \\9 - ( x + 2 )\end{array} \right) } { \left( \begin{array} { c } 18 \\9\end{array} \right) } & x = 0,1 \ldots , 7 \\0 & \text { otherwise. }\end{array} \right.$

Question 4

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.

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.

Question 6

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}?

Question 7

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.

Question 8

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.

Question 9

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).

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.

Question 11

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?

Question 12

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).

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.

Question 14

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