# Exam P Practice Problem 59 – Joint Distributions

Problem 59-A

Two random losses $X$ and $Y$ are jointly modeled by the following density function:

$\displaystyle f(x,y)=\frac{1}{32} \ (4-x) \ (4-y) \ \ \ \ \ \ 0

Suppose that both of these losses had occurred. Given that $X$ exceeds 2, what is the probability that $Y$ is less than 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.4000$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.4667$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.7518$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8571$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.9375$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

Problem 59-B

Two random losses $X$ and $Y$ are jointly modeled by the following density function:

$\displaystyle f(x,y)=\frac{1}{96} \ (x+2y) \ \ \ \ \ \ 0

Suppose that both of these losses had occurred. What is the probability that only one of them exceeds 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.1250$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.2083$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.2917$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.3750$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.5000$

____________________________________________________

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

____________________________________________________

____________________________________________________

$\copyright \ 2013$

# Exam P Practice Problem 32 – Covariance

Problem 32A

Suppose that $X$ and $Y$ are two random losses with the following joint density function:

$\displaystyle f(x,y)=\frac{1}{24} \ (x+y) \ \ \ \ \ 0

Calculate the covariance of these two losses.

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

Problem 32B

Suppose that $X$ and $Y$ are two random losses with the following joint density function:

$\displaystyle f(x,y)=\frac{1}{8} \ (x+y) \ \ \ \ \ 0

Calculate the covariance of these two losses.

_______________________________________________________

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

_______________________________________________________

_______________________________________________________

$\copyright \ 2013$

# Exam P Practice Problem 31 – Covariance

Problem 31A

A traveler is at the airport of City A and wants to reach his home in City B. He plans to take an airplane flight from City A to the airport of City B. Once he arrives in the airport of City B, he plans to reach his home by riding in a bus. Let $X$ be the time (in hours) of his flight from City A to City B. Let $Y$ be the time (in hours) of the bus ride. The following is the joint density function of $X$ and $Y$.

$\displaystyle f(x,y)=\frac{1}{16} \ x \ y \ \ \ \ \ \ \ 0

Let $W$ be the total time of travel from the airport of City A to his home in City B. Calculate the covariance of $X$ and $W$.

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

Problem 31B

An investor is facing two potential financial losses $X$ and $Y$ with the following joint density function:

$\displaystyle f(x,y)=0.025 \ x \ e^{-0.2 \ y} \ \ \ \ \ \ \ 0

Let $T$ be the total of these two losses. Calculate the covariance of $T$ and $Y$.

___________________________________________________________

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

___________________________________________________________

$\copyright \ 2013$