probability gamma distribution

[logistic_guy]

here is the question

For an electrical component with a failure rate of once every \(\displaystyle 5\) hours, it is important to consider the time that it takes for \(\displaystyle 2\) components to fail.

(a) Assuming that the gamma distribution applies, what is the mean time that it takes for \(\displaystyle 2\) components to fail?
(b) What is the probability that \(\displaystyle 12\) hours will elapse before \(\displaystyle 2\) components fail?


my attemv
the PDF of gamma distribution \(\displaystyle f(x) = \frac{x^{\alpha - 1}}{\Gamma(\alpha)\beta^{\alpha}}e^{-\frac{x}{\beta}}\)
the mean \(\displaystyle \alpha \beta\)
how to tell what's \(\displaystyle \alpha\) and \(\displaystyle \beta\) from the question?
it's also confusing for (b)
do the probability \(\displaystyle P(X > 12)\) or \(\displaystyle P(X < 12)\)
it's confusing

 

[khansaheb]

how to tell what's α\displaystyle \alphaα and β\displaystyle \betaβ from the question?

Read TEXTBOOK and/or do research with GOOGLE!!

 

[simba087]

Dating for Sex.

 

[lookagain]

Post #3 and another one like it in a different thread were reported to be asked for deletion and permanent banning. If no one on this site can make that happen, then this site is in more trouble than I had thought.

 

[jonah2.0]

Beer drenched reaction follows.

Post #3 and another one like it in a different thread were reported to be asked for deletion and permanent banning. If no one on this site can make that happen, then this site is in more trouble than I had thought.

Management thanks you profusely for your comments, Inspector lookagain. Rest assured that such will be duly noted in your Personnel (yes, 2 n) File, and seriously taken in consideration at your forthcoming Annual Performance Review (not to be confused with the APR associated with financial jargon).

Cheers.

 

[logistic_guy]

In this type of problems, always look for the rate, \(\displaystyle \lambda\).

\(\displaystyle \lambda = \frac{1}{5}\)

\(\displaystyle \beta\) is the inverse of \(\displaystyle \lambda\), so:

\(\displaystyle \beta = \frac{1}{\lambda} = \frac{1}{\frac{1}{5}} = 5\)

Then, the other parameter \(\displaystyle 2\) must be \(\displaystyle \alpha\) as gamma distribution depends on two parameters, \(\displaystyle \alpha\) and \(\displaystyle \beta\).

\(\displaystyle \alpha = 2\)

And the mean time is:

\(\displaystyle E(x) = \alpha\beta = 2(5) = 10 \ \text{hours}\)

 

[logistic_guy]

\(\displaystyle \bold{(b)}\)

\(\displaystyle P(X \geq 12) = \frac{1}{\beta^2\Gamma{(\alpha)}}\int_{12}^{\infty}xe^{-\frac{x}{\beta}} \ dx = \frac{1}{5^2\Gamma{(2)}}\int_{12}^{\infty}xe^{-\frac{x}{5}} \ dx \approx 0.308441\)