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]

$\displaystyle \bold{(b)}$

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