component failing the gamma distribution Suppose the time in days until a component fails has the gamma distribution with α = 5 and θ = 1/10. When
a component fails, it is immediately replaced by a new component. Use the central limit theorem to estimate the
probability that 40 components will together be suﬃcient to last at least 6 years. (Assume that a year has 365.25
days.)
central limit theorem :Let X1, X2,... be i.i.d with ﬁnite mean µ and ﬁnite variance σ
2
, let Z_n
=(X_n−µ)/
(σ/
√
n)
and
Z ∼ N(0,1). Then Z_
n
−→ Z.
 A: You have listed  all the necessary ingredients. Now we need to put them together.
Let random variable $X_1$ be the length of working life of the first component, once it is put into action. Similarly, let $X_2$ be the length of working life of the second component, and so on up to $X_{40}$, the length of working life of the $40$-th component. 
You want to estimate the probability that $Y\ge 6(365.25)$, where
$$Y=X_1+X_2+\cdots +X_{40}.$$
So find the mean $\mu$ and variance $\sigma^2$ of each $X_i$. You can look up this information, or compute it. 
Then the mean of $Y$ is $40\mu$, and the variance of $Y$ is $40\sigma^2$. 
The random variable $Y$ is a sum of a not too small number of independent identically distributed random variables with a respectable distribution. So to compute probabilities with reasonble accuracy, we can pretend that $Y$ has normal distribution (it doesn't). 
Calculate in the usual way the probability that a normal with mean $40\mu$ and variance $40\sigma^2$ is $\ge 6(365.25)$. 
