# gamma distribution

• Feb 21st 2009, 07:07 PM
Yan
gamma distribution
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? • Feb 21st 2009, 07:33 PM matheagle 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. • Feb 22nd 2009, 09:38 AM Yan the question is <how many salespersons should the company employ to maximize the expected profit?> that is why i don't know hoe to do it. can anyone help me? I'm really needs help!!! • Feb 22nd 2009, 11:03 AM DrewTV 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!
-Drew
• Feb 22nd 2009, 11:11 AM
Moo
Quote:

Originally Posted by Yan
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 $8,000 per salesperson, how many salespersons should the company employ to maximize the expected profit? As matheagle said, the mean of a gamma distribution is$\displaystyle \alpha \beta=160 \sqrt{n}$The global sales cost is 8n (we're counting in thousands of dollars) The expected profit is 160*sqrt(n) - 8n Then maximize this function with respect to n. edit : just a little too late ^^' • Feb 22nd 2009, 01:56 PM Yan Quote: Originally Posted by Moo As matheagle said, the mean of a gamma distribution is$\displaystyle \alpha \beta=160 \sqrt{n}\$
The global sales cost is 8n (we're counting in thousands of dollars)

The expected profit is 160*sqrt(n) - 8n

Then maximize this function with respect to n.

edit : just a little too late ^^'

So, I just need to find the derivative of the net profit?
• Feb 22nd 2009, 02:00 PM
Moo
Quote:

Originally Posted by Yan
So, I just need to find the derivative of the net profit?

Yes :), like DrewTV said.