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

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

\text{ }

\text{ }

\text{ }

\text{ }

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<x<\infty, \ \ \ 0<y<5

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

__________________________________________________

\text{ }

\text{ }

\text{ }

\text{ }

\text{ }

\text{ }

\text{ }

\text{ }

__________________________________________________

Answers

__________________________________________________

\copyright \ 2013

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: