Exam 9: Multivariate Distributions

For what value of is $f ( x , y , z ) = \left\{ \begin{array} { l l } \frac { 1 } { c } e ^ { x + y + z } & x , y , z \leq 2 \\0 & \text { else. }\end{array} \right.$ a joint probability density function? For that value of c, find P(X

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Question 1

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c=e⁶ and P(X

A drawer contains 10 white socks, 14 blue socks, and 8 green socks. Suppose that 9 socks are chosen at random from the drawer, let X be the number of white socks, Y the number of blue socks, and Z the number of green socks chosen. Give the joint probability mass function for X, Y, and Z.

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Question 2

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$p ( x , y , z ) = \left\{ \begin{array} { l l } \frac { \left( \begin{array} { c } ( 410 \\x\end{array} \right) \left( \begin{array} { l } 14 \\y\end{array} \right) \left( \begin{array} { l } 8 \\z\end{array} \right) } { \left( \begin{array} { l } 32 \\9\end{array} \right) } & x = 0 , \ldots , 9 ; y = 0 , \ldots , 9 ; z = 0 , \ldots 8 ; x + y + z = 9 \\0 & \text { elsewhere }\end{array} \right.$

A random integer from 1 to 5 (inclusive) is chosen at random 12 times (with replacement). What is the probability that 1, 2, 4 and 5 appear 3 times each?

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Question 3

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$\frac { 12 ! } { 3 ! 3 ! 3 ! 3 ! } \left( \frac { 1 } { 5 } \right) ^ { 12 } = .0015$

At a certain Kennel, 25% of the dogs admitted are Spaniels, %30 are Labradors, %24 are Collies, and the remainder are Mastiffs. If 25 dogs are randomly selected and let out to play, find the probability that 10 are Labradors, 4 are Collies, and at most 2 are Mastiffs.

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Question 4

SSuppose that you choose a point at random in the tetrahedron formed by the planes x=0, y=0, z=0, and x+y+z=1. Let X,Y,Z represent the x-, y-, and z-coordinates of that point, respectively. (a) Find the joint probability density function of X, Y and Z. (b) Find the probability density function of Z marginalized over X and Y.

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Question 5

Let X,Y ,Z and be random variables with joint probability density function $f ( x ) = \left\{ \begin{array} { l l } 8 x y z & 0 < x < 1,0 < y < 1,0 < z < 1 \\0 & \text { else. }\end{array} \right.$ Find P(Z

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Question 6

A certain sandwich shop has a cooler full of sandwiches. Suppose that 45% of its sandwiches are turkey, 25% are ham, and 30% are vegetable sandwiches. A customer comes in and grabs 14 sandwiches randomly. What is the probability that she gets exactly 7 turkey and at most 3 vegetable sandwiches?

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Question 7

A restaurant is about to close, but it must wait until all 8 remaining customers have left. Starting now, each customer's remaining time in the restaurant is exponentially distributed with mean fifteen minutes. If the customers leave independently of one another, find the probability that (a) after 2 hours the restaurant is still open; (b) after 1 hour, the restaurant is still waiting on exactly 2 customers.

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Question 8

A certain pool is staffed by unreliable lifeguards. The lifeguards are numbered 1 through n and they all start on duty at the same time every day. The pool remains open as long as at least two of the lifeguards are on duty. Suppose that, on any day, independently of the others, lifeguard i stops working at a time which is a random variable with distribution function Fᵢ. In terms of the Fᵢ's, find the survival function of the period that the pool stays open.

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Question 9

Suppose that 70% of a certain car manufacturer's production are sedans, 20% are SUVs, and 10% are trucks. If 17 vehicles of this manufacturer are selected randomly, what is the probability of exactly 9 sedans and exactly 1 truck or exactly 8 sedans and exactly 2 trucks?

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Question 10

Three radio stations decide to play the same song independently of one another. They each start playing the song randomly sometime between 12:00pm and 12:25pm. (a) Find the probability that the song plays for the last time between 12:18pm and 12:22pm. (b) Find the probability that all three are played between 12:05pm and 12:20pm.

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Question 11

The random variable X, Y and Z have joint probability density function $f ( x , y , z ) = \left\{ \begin{array} { l l } \frac { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) } { K } & 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \\0 & \text { else }\end{array} \right.$ (a) Find K. (b) Find P(X>.2|Y+Z>1).

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Question 12

You are making a Jello mold in the shape of an inverted right cone, radius 3cm at the bottom and 6 cm high. You place 3 cherries in the Jello for your guests to find. They end up distributed randomly in the Jello. Find the probability that none of the Cherries end up within 1cm of the exterior surface of the cone.

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Question 13

You have two bank accounts, and you do not know what the balance of each account is. All you remember is that each account's balance is a random amount between $10 and $200. Find the expected value of the difference between the balances of the two accounts.

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Question 14

For what value of is $f ( x , y , z ) = \left\{ \begin{array} { l l } \frac { 1 } { c } e ^ { x + y + z } & x , y , z \leq 2 \\0 & \text { else. }\end{array} \right.$ a joint probability density function? For that value of c, find P(X

A drawer contains 10 white socks, 14 blue socks, and 8 green socks. Suppose that 9 socks are chosen at random from the drawer, let X be the number of white socks, Y the number of blue socks, and Z the number of green socks chosen. Give the joint probability mass function for X, Y, and Z.

A random integer from 1 to 5 (inclusive) is chosen at random 12 times (with replacement). What is the probability that 1, 2, 4 and 5 appear 3 times each?

At a certain Kennel, 25% of the dogs admitted are Spaniels, %30 are Labradors, %24 are Collies, and the remainder are Mastiffs. If 25 dogs are randomly selected and let out to play, find the probability that 10 are Labradors, 4 are Collies, and at most 2 are Mastiffs.

Let X,Y ,Z and be random variables with joint probability density function $f ( x ) = \left\{ \begin{array} { l l } 8 x y z & 0 < x < 1,0 < y < 1,0 < z < 1 \\0 & \text { else. }\end{array} \right.$ Find P(Z

A certain sandwich shop has a cooler full of sandwiches. Suppose that 45% of its sandwiches are turkey, 25% are ham, and 30% are vegetable sandwiches. A customer comes in and grabs 14 sandwiches randomly. What is the probability that she gets exactly 7 turkey and at most 3 vegetable sandwiches?

Suppose that 70% of a certain car manufacturer's production are sedans, 20% are SUVs, and 10% are trucks. If 17 vehicles of this manufacturer are selected randomly, what is the probability of exactly 9 sedans and exactly 1 truck or exactly 8 sedans and exactly 2 trucks?

You have two bank accounts, and you do not know what the balance of each account is. All you remember is that each account's balance is a random amount between $10 and $200. Find the expected value of the difference between the balances of the two accounts.

Axioms of Peobability

Combinatorial Metods

Conditional Probability and Independence

Distribution Functions and Discrete Random Variables

Apecial Discrete Distributions

Continuous Random Variables

Special Continuous Distribution

Bivariate Distributions

More Expectations and Variances

Sums of Independent Random Variables and Limit Theorems

Stocastic Processes

Filters

Exam 9: Multivariate Distributions

A drawer contains 10 white socks, 14 blue socks, and 8 green socks. Suppose that 9 socks are chosen at random from the drawer, let X be the number of white socks, Y the number of blue socks, and Z the number of green socks chosen. Give the joint probability mass function for X, Y, and Z.

A random integer from 1 to 5 (inclusive) is chosen at random 12 times (with replacement). What is the probability that 1, 2, 4 and 5 appear 3 times each?

At a certain Kennel, 25% of the dogs admitted are Spaniels, %30 are Labradors, %24 are Collies, and the remainder are Mastiffs. If 25 dogs are randomly selected and let out to play, find the probability that 10 are Labradors, 4 are Collies, and at most 2 are Mastiffs.

Let X,Y ,Z and be random variables with joint probability density function f(x)={8xyz0<x<1,0<y<1,0<z<10 else. f ( x ) = \left\{ \begin{array} { l l } 8 x y z & 0 < x < 1,0 < y < 1,0 < z < 1 \\0 & \text { else. }\end{array} \right.f(x)={8xyz0​0<x<1,0<y<1,0<z<1 else. ​ Find P(Z

A certain sandwich shop has a cooler full of sandwiches. Suppose that 45% of its sandwiches are turkey, 25% are ham, and 30% are vegetable sandwiches. A customer comes in and grabs 14 sandwiches randomly. What is the probability that she gets exactly 7 turkey and at most 3 vegetable sandwiches?

Suppose that 70% of a certain car manufacturer's production are sedans, 20% are SUVs, and 10% are trucks. If 17 vehicles of this manufacturer are selected randomly, what is the probability of exactly 9 sedans and exactly 1 truck or exactly 8 sedans and exactly 2 trucks?

You have two bank accounts, and you do not know what the balance of each account is. All you remember is that each account's balance is a random amount between $10 and $200. Find the expected value of the difference between the balances of the two accounts.

Axioms of Peobability

Combinatorial Metods

Conditional Probability and Independence

Distribution Functions and Discrete Random Variables

Apecial Discrete Distributions

Continuous Random Variables

Special Continuous Distribution

Bivariate Distributions

More Expectations and Variances

Sums of Independent Random Variables and Limit Theorems

Stocastic Processes

Filters

Let X,Y ,Z and be random variables with joint probability density function $f ( x ) = \left\{ \begin{array} { l l } 8 x y z & 0 < x < 1,0 < y < 1,0 < z < 1 \\0 & \text { else. }\end{array} \right.$ Find P(Z