Continuous RV

Danneedshelp · #1 $Old$ October 26th, 2009, 03:14 PM

Q: The lifetime (in hours) Y of an electronic component isa random variable with density function given by

$f(y)=\left\{\begin{array}{cc}\frac{1}{100}e^{\frac{-y}{100}},&\mbox{}y>0\\0,&\mbox{}elsewhere\end{array}\right.$

Three of these components operate independently in a piece of equipment. The equipment fails if at least two of the components fail. Find the probability that the equipment will operate for at least 200 hours without failure.

I need help breaking this problem down into its peices. Clearly, $Y\sim\\EXP(1,100)$ , but I am not sure how to attack the first part of the problem, "three of these components operate independently in a piece of equipment. The equipment fails if at least two of the components fail". So, do I have to calculate the probability that the equipment works fisrt and then multiply that that by the probability the equipment will operate at least 200 hours?

To find the probabilty of the equipment working, I just need to find 1-two out of the three components work. Can I use the poisson distribtuion with $\lambda=\frac{1}{100}$ to compute $1-P(Y\geq\\2)$ ?

Another questions: what is the relationship between $\lambda$ and $\beta$ when dealing with the exponential sub family of the gamma distribution?

artvandalay11 · #2 $Old$ October 26th, 2009, 07:45 PM

Quote:

Originally Posted by Danneedshelp $View Post$

Q: The lifetime (in hours) Y of an electronic component isa random variable with density function given by

$f(y)=\left\{\begin{array}{cc}\frac{1}{100}e^{\frac{-y}{100}},&\mbox{}y>0\\0,&\mbox{}elsewhere\end{array}\right.$

Three of these components operate independently in a piece of equipment. The equipment fails if at least two of the components fail. Find the probability that the equipment will operate for at least 200 hours without failure.

I need help breaking this problem down into its peices. Clearly, $Y\sim\\EXP(1,100)$ , but I am not sure how to attack the first part of the problem, "three of these components operate independently in a piece of equipment. The equipment fails if at least two of the components fail". So, do I have to calculate the probability that the equipment works fisrt and then multiply that that by the probability the equipment will operate at least 200 hours?

To find the probabilty of the equipment working, I just need to find 1-two out of the three components work. Can I use the poisson distribtuion with $\lambda=\frac{1}{100}$ to compute $1-P(Y\geq\\2)$ ?

Another questions: what is the relationship between $\lambda$ and $\beta$ when dealing with the exponential sub family of the gamma distribution?

This problem is straight out of Wackerly's book and yes, find the probability of one failing, using integration, then use this as the probability for binomial distribution with n=3