9.2 Problems on Independent Classes of Random Variables

Testing whether X and Y are independent from a joint density supported on a square, circle or triangle, and exploiting independence where it holds.

Applied Probability, 9.2 Problems on Independent Classes of Random Variables

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22 exercises

9.2 Exercise 4

$f_{X Y} \left(t , u\right) = 1 / \pi$ on the circle with radius one, center at (0,0).

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9.2 Exercise 5

$f_{X Y} \left(t , u\right) = 1 / 2$ on the square with vertices at $\left(1 , 0\right) , \left(2 , 1\right) , \left(1 , 2\right) , \left(0 , 1\right)$ (see Exercise 11 from "Problems on Random Vectors and Joint Distributions").

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9.2 Exercise 6

$f_{X Y} \left(t , u\right) = 4 t \left(1 - u\right)$ for $0 \leq t \leq 1$ , $0 \leq u \leq 1$ (see Exercise 12 from "Problems on Random Vectors and Joint Distributions").

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9.2 Exercise 7

$f_{X Y} \left(t , u\right) = \frac{1}{8} \left(t + u\right)$ for $0 \leq t \leq 2$ , $0 \leq u \leq 2$ (see Exercise 13 from "Problems on Random Vectors and Joint Distributions").

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9.2 Exercise 8

$f_{X Y} \left(t , u\right) = 4 u e^{- 2 t}$ for $0 \leq t , 0 \leq u \leq 1$ (see Exercise 14 from "Problems on Random Vectors and Joint Distributions").

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9.2 Exercise 9

$f_{X Y} \left(t , u\right) = 12 t^{2} u$ on the parallelogram with vertices $\left(- 1 , 0\right) , \left(0 , 0\right) , \left(1 , 1\right) , \left(0 , 1\right)$

(see Exercise 16 from "Problems on Random Vectors and Joint Distributions").

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9.2 Exercise 10

$f_{X Y} \left(t , u\right) = \frac{24}{11} t u$ for $0 \leq t \leq 2$ , $0 \leq u \leq \min \left\{1 , 2 - t\right\}$ (see Exercise 17 from "Problems on Random Vectors and Joint Distributions").

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9.2 Exercise 11

Two software companies, MicroWare and BusiCorp, are preparing a new business package in time for a computer trade show 180 days in the future. They work independently. MicroWare has anticipated completion time, in days, exponential (1/150). BusiCorp has time to completion, in days, exponential (1/130). What is the probability both will complete on time; that at least one will complete on time; tha…

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9.2 Exercise 12

Eight similar units are put into operation at a given time. The time to failure (in hours) of each unit is exponential (1/750). If the units fail independently, what is the probability that five or more units will be operating at the end of 500 hours?

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9.2 Exercise 13

The location of ten points along a line may be considered iid random variables with symmytric triangular distribution on $\left[1 , 3\right]$ . What is the probability that three or more will lie within distance 1/2 of the point $t = 2$ ?

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9.2 Exercise 14

A Christmas display has 200 lights. The times to failure are iid, exponential (1/10000). The display is on continuously for 750 hours (approximately one month). Determine the probability the number of lights which survive the entire period is at least 175, 180, 185, 190.

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9.2 Exercise 15

A critical module in a network server has time to failure (in hours of machine time) exponential (1/3000). The machine operates continuously, except for brief times for maintenance or repair. The module is replaced routinely every 30 days (720 hours), unless failure occurs. If successive units fail independently, what is the probability of no breakdown due to the module for one year?

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9.2 Exercise 16

Joan is trying to decide which of two sales opportunities to take.

In the first, she makes three independent calls. Payoffs are $570, $525, and $465, with respective probabilities of 0.57, 0.41, and 0.35.
In the second, she makes eight independent calls, with probability of success on each call $p = 0 . 57$ . She realizes $150 profit on each successful sale.

Let X be the net profit on the first alterna…

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9.2 Exercise 17

Margaret considers five purchases in the amounts 5, 17, 21, 8, 15 dollars with respective probabilities 0.37, 0.22, 0.38, 0.81, 0.63. Anne contemplates six purchases in the amounts 8, 15, 12, 18, 15, 12 dollars. with respective probabilities 0.77, 0.52, 0.23, 0.41, 0.83, 0.58. Assume that all eleven possible purchases form an independent class.

What is the probability Anne spends at least twice as…

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9.2 Exercise 18

James is trying to decide which of two sales opportunities to take.

In the first, he makes three independent calls. Payoffs are $310, $380, and $350, with respective probabilities of 0.35, 0.41, and 0.57.
In the second, he makes eight independent calls, with probability of success on each call $p = 0 . 57$ . He realizes $100 profit on each successful sale.

Let X be the net profit on the first alternative…

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9.2 Exercise 21

The class $\left\{A , B , C , D , E , F\right\}$ is independent; the respective probabilites for these events are $\left\{0 . 46 , 0 . 27 , 0 . 33 , 0 . 47 , 0 . 37 , 0 . 41\right\}$ . Consider the simple random variables

X = 3 I_{A} - 9 I_{B} + 4 I_{C} , Y = - 2 I_{D} + 6 I_{E} + 2 I_{F} - 3 , \text{and} Z = 2 X - 3 Y

Determine $P \left(Y > X\right)$ , $P \left(Z > 0\right)$ , $P \left(5 \leq Z \leq 25\right)$ .

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9.2 Exercise 22

Two players, Ronald and Mike, throw a pair of dice 30 times each. What is the probability Mike throws more “sevens” than does Ronald?

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9.2 Exercise 23

A class has fifteen boys and fifteen girls. They pair up and each tosses a coin 20 times. What is the probability that at least eight girls throw more heads than their partners?

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9.2 Exercise 24

Glenn makes five sales calls, with probabilities $0.37$ , $0.52$ , $0.48$ , $0.71$ , $0.63$ , of success on the respective calls. Margaret makes four sales calls with probabilities $0.77$ , $0.82$ , $0.75$ , $0.91$ , of success on the respective calls. Assume that all nine events form an independent class. If Glenn realizes a profit of $18.00 on each sale and Margaret earns $20.00 on each sale, what is the probability Margaret's gain is at least $10…

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9.2 Exercise 25

Mike and Harry have a basketball shooting contest.

Mike shoots 10 ordinary free throws, worth two points each, with probability 0.75 of success on each shot.
Harry shoots 12 “three point” shots, with probability 0.40 of success on each shot.

Let $X , Y$ be the number of points scored by Mike and Harry, respectively. Determine $P \left(X \geq 15\right)$ , and $P \left(Y \geq 15\right) , P \left(X \geq Y\right)$ .

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9.2 Exercise 26

Martha has the choice of two games.

Game 1Pay ten dollars for each “play.” If she wins, she receives $20, for a net gain of $10 on the play; otherwise, she loses her $10. The probability of a win is 1/2, so the game is “fair.”
Game 2Pay five dollars to play; receive $15 for a win. The probability of a win on any play is 1/3.

Martha has $100 to bet. She is trying to decide whether to play Game…

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9.2 Exercise 27

Jim and Bill of the men's basketball team challenge women players Mary and Ellen to a free throw contest. Each takes five free throws. Make the usual independence assumptions. Jim, Bill, Mary, and Ellen have respective probabilities $p = 0.82 , 0.87 , 0.80$ , and 0.85 of making each shot tried. What is the probability Mary and Ellen make a total number of free throws at least as great as the total made by the guys?

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