probability gamma distribution

logistic_guy

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here is the question

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

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


my attemv
the PDF of gamma distribution f(x)=xα1Γ(α)βαexβ\displaystyle f(x) = \frac{x^{\alpha - 1}}{\Gamma(\alpha)\beta^{\alpha}}e^{-\frac{x}{\beta}}
the mean αβ\displaystyle \alpha \beta
how to tell what's α\displaystyle \alpha and β\displaystyle \beta from the question?
it's also confusing for (b)☹️
do the probability P(X>12)\displaystyle P(X > 12) or P(X<12)\displaystyle P(X < 12)
it's confusing🥺
 
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.
 
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.
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Cheers.
 
In this type of problems, always look for the rate, λ\displaystyle \lambda.

λ=15\displaystyle \lambda = \frac{1}{5}

β\displaystyle \beta is the inverse of λ\displaystyle \lambda, so:

β=1λ=115=5\displaystyle \beta = \frac{1}{\lambda} = \frac{1}{\frac{1}{5}} = 5

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

α=2\displaystyle \alpha = 2

And the mean time is:

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

P(X12)=1β2Γ(α)12xexβ dx=152Γ(2)12xex5 dx0.308441\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
 
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