Question: Suppose the time, in days, until a component fails has the Gamma distribution with alpha=5 and lambda=1/10. When a component fails it is immediately replaced

Suppose the time, in days, until a component fails has the Gamma distribution with alpha=5 and lambda=1/10. When a component fails it is immediately replaced by a new component. Determine the exact distribution of the total time that the 40 components will last. Then use the function pgamma in R to find the exact probability and compare it to the normal approximations.

I get the probability using CLT and the gamma distribution using mgf. But I'm not sure how to find the exact probability using pgamma function in R. Thanks.

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