The diameter X of a manufactured part is an exponential rand

The diameter X of a manufactured part is an exponential random variable with parameter beta. If X 1 the part must be discarded at a net loss of $1. The machinery manufacturing the part may be set so that beta = 4 or beta = 2. Which setting will maximize the manufacturer\'s expected profit?

Solution

Let X has exponential distribution with parameter beta... that means X has mean 1/beta

we want to max probability in {x<1}

for beta=2

P(X<1)=0.864665

for beta =4

P(X<1)=.981684

hence we choose beta=4