Lets say that there's a device, which contains 2 consumable components. When the first component fails, the second is started automatically. When both have failed, someone has to change the components. All components are the same. Their uptime follows a gamma distribution with:
Mean $\mu$ = 2700 hours
Sigma $\sigma$ = 400 hours
What's the probability that both components break at earliest after 6000 hours?
My attempt: I thought that this must be solved by using the Erlang distribution (how many events occur in the given time). I tried converting the parameters from gamma to erlang format:
Variance $\sigma^2 = (400)^2 = 160000$ hours ??
$\mu = \alpha\beta$
$\alpha = \mu/\beta$
$\sigma^2=\alpha\beta^2$
$(\mu/\beta)\beta^2=\mu\beta$
$\beta = \sigma^2/\mu = 400^2/2700 = 1600/27$
According to gamma formula, the value of $\alpha=\mu/\beta$, but in Erlang we should use $\alpha$=n=2 (count of broken components), right?
Y ~ erlang(t; n, β) ~ gamma(t; n, β)
P(Y > 6000)
= 1 - P(Y <= 6000)
= 1 - Gamma(6000; n, β)
= 1 - gamcdf(6000, n, β) // Run in Matlab
= 1 - gamcdf(6000, 2, (1600/27))
This however, results in probability of 0. Is this the correct way to calculate the probability?