Exam 10: More Expectations and Variances

Let X, Y, and Z be random variables with joint probability density function $f ( x , y , z ) = \left\{ \begin{array} { l l } 54 x ^ { 2 } y ^ { 2 } z ^ { 2 } & 0 < y , z < 1,0 < x < y \\0 & \text { else. }\end{array} \right.$ Find ρ(X,Y).

Let X and Y have joint probability density function $\left\{ \begin{array} { l l } \frac { x + 2 y } { 16 } & 1 < x < 3,0 < y < 2 \\0 & \text { else }\end{array} \right.$ Find E(√X+x³ Y²).

There are two gas stations at a busy intersection. The Kwick Pump and the Gas Hog. Customers arrive at the Kwick pump according to a Poisson process with rate 2 per minute and customers arrive to the Gas Hog according to a Poisson process with rate 3 per minute. Let N be the number of customers arriving to the Gass Hog during the time that 4 customers arrive to the Kwick Pump. Find the expected value of N.

A dating app is meant to arrange dates between 13 dog lovers and 13 dog owners, each dog lover is matched with one dog owner. However, due to a glitch, instead of sending dog lovers to the matched dog owners, the app sends the dog lovers to the houses of dog owners randomly. Find the expected number of dog lovers who end up at the correct home.

A certain fishing vessel only keeps tuna that weigh above 50 pounds. For each tuna that they sell, they receive an amount equal to the weight of the tuna, in dollars, up to a maximum of $80. For a tuna that weighs above 80 pounds, they still receive $80. If the weight of a random tuna the vessel keeps is 50+X, where X is an exponential random variable with mean 25 pounds, find the expected amount the vessel receives for a random tuna.

You roll a die 5 times. Let $X _ { i } = \left\{ \begin{array} { l l } 1 & \text { the } i ^ { \text {th } } \text { roll is even } \\0 & \text { elsewhere }\end{array} \right.$ For 1 ≤ i≤ j≤ 5 find Cov(Xᵢ,Xⱼ).

Suppose that X, Y and Z are random variables with standard deviations σₓ=1, σᵧ=2 and σz=3 and that ρ(X,Y)=.5, ρ(X,Z)=-.2 and ρ(Y,Z)=.2. Find Var(X+Y+Z).

A random point (X,Y) is chosen at random from the rectangle [0,2]×[0,2]. Find P(X

A custodian has 9 identical keys of which only one opens room number 102's door. However, she has forgotten which one. If she tries them randomly one at a time and without replacement, find the expected number of keys she must try before getting the correct one.

Let X be randomly distributed with probability density function $f ( x ) = \left\{ \begin{array} { l l } \frac { x ^ { 2 } + 2 x } { 60 } & 2 \leq x \leq 5 \\0 & \text { else }\end{array} \right.$ Find $E \left( \frac { 1 } { x } \right)$

Pulling trailers reduce the gas mileage. Suppose that an unloaded car gets X miles per gallon, and a car towing a trailer gets Y miles per gallon, where X and Y are random variables with a joint density which is bivariate normal with μx=22, σx=2.2, μy=14, σy=3.2 and ρ=.4. If a certain car gets 24 miles per gallon without a trailer, find the probability that it gets at most 15 miles per gallon with a trailer.

Two students are walking their dogs around a park. Mary's dog sniffs a tree at a Poisson rate of 4 times per hour, while Antoine's dog sniffs a tree at a Poisson rate of 3 times per hour independently of Mary's dog. If Mary's dog sniffed the last 2 trees, what is the probability that Antoine's dog will sniff a tree next?

You are rolling a fair 20-sided die. Find the expected number of rolls you need to toss to get 3 consecutive rolls of outcomes 13 or above.

Suppose that a company produces snow globes 24 hours a day, 7 days a week. Furthermore, suppose that some of the snow globes are defective and the average number of defective snow globes produced per day is a gamma random variable with parameters 4 and λ=2. If the defective globes are produced according to a Poisson process, find the probability density function of the average number of defective snow globes after a week.