Suppose the time in days until a component fails has the gamma distribution with alpha=5 and theta = 1/10. When a component fails, it gets immediately replaced by a new one. 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 is 365.25 days.

My attempt

E(X)=0.5=mew=sample mew

Var(x)=1/20

sigma = sqrt(1/20)

sample sigma [40]= sqrt([1/20] / 40) = 0.035

so then

= 1 - P(sample X[40] <= 6*365.25)

= 1 - P(sample X[40] <= 2191.5)

= 1 - P([sample X[40] - 0.5 ] / 0.035 <= [2191.5-0.5]/0.035)

= 1 - P(X<=62600) where X~N(0,1)

~0