I don't really understand the question.
The mean of a gamma is alpha times beta, but I don't know what's going on here.
If a company employs n salespersons, its gross sales in thousands of dollars may be regarded as a random variable having a gamma distribution with α=80(n)^(1/2) and β=2. If the sales cost $8,000 per salesperson, how many salespersons should the company employ to maximize the expected profit?
I am also working on this problem and I think I figured it out.
Expected profit in this question means net profit, so
Expected= net = gross profit - cost
Gross profit is the gamma distribution with given alpha and beta, and we know the expectation of the gamma distribution is alpha times beta.
Cost is $8000n where n is the number of employees, but we must remember that the gamma distribution is given in thousands of dollars, so we adjust the cost to be 8n (in thousands of dollars)
Now the net profit is a function of n:
Y = (80√n)(2) – 8n
And this curve has a maximum where it's derivative is 0, so take the derivative, set it equal to zero and solve for n.
I hope this helps!