# Exam P Practice Problem 57 – Lifetimes of Machines

Problem 57-A

A factory owner purchased two identical machines for her factory. Let $X$ and $Y$ be the lifetimes (in years) of these two machines. These lifetimes are modeled by the following joint probability density function.

$\displaystyle f(x,y)=\frac{0.01}{\sqrt{x} \ \sqrt{y}} \ e^{-0.2 \sqrt{x}} \ e^{-0.2 \sqrt{y}} \ \ \ \ \ \ \ 0

The machine whose lifetime is modeled by the random variable $Y$ came online 2 years after the beginning of operation of the machine that is modeled by the random variable $X$.

Given that $X$ exceeds 2, that is the probability that $Y$ exceeds 3?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.2928$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.4670$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.5330$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 0.7072$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 0.7536$

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Problem 57-B

A company purchased two machines for its factory. Let $X$ and $Y$ be the lifetimes (in years) of these machines. The following is the joint density function of their lifetimes.

$\displaystyle f(x,y)=\frac{3}{125} \ y \ e^{-0.3 x} \ \ \ \ \ \ \ 0

The machine whose lifetime is modeled by the random variable $Y$ came online after the failure of the machine whose lifetime is modeled by $X$.

What is the variance of the total time of operation of these two machines?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 12.50$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 13.60$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 17.20$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 19.85$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 23.61$

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