Tag Archives: Covariance

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<x<4, \ 0<y<x

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

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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<x<4, \ 0<y<4

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

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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<x<4, \ \ 0<y<2

Calculate the covariance of these two losses.

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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<x<2, \ \ 0<y<2

Calculate the covariance of these two losses.

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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<x<4, \ \ \  0<y<2

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.

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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<x<4, \ \ \  0<y<\infty

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

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