Uniform probability density function question

[1daj]

1. Homework Statement

A vendor at a market buys mushrooms from a wholesaler for $3 a pound, and sells them for $4 a pound. The daily demand (in pounds) from custumers for the vendor;s mushrooms is a random variable X with pdf
f(x) = 1/40 if 0 (greater than) x (less than) 40 and 0 elsewhere
Unsold mushrooms must be thrown out at the end of the day.
Suppose that the vendor buys a constant amount, say C pounds, of mushrooms from the wholesaler at the beginning of each day (0 (less than) C (greater than) 40). Find the expected value of the vendor's daily profit.

Homework Equations

The Attempt at a Solution

I am really confused as to how to set up the solution to this problem and this is how i set it up:

E(profit) = 4 E(demand) - 3C = 4 (integral from 0 to c) x/40 dx - 3C
= 4 (c^2)/80 - 3C
= (c^2)/20 -3C
What troubles me about this solution is that the expected profit is never positive. Furthermore I think my problem is with finding the the Expected demand
Any help is greatly appreciated.

 

[Dick]

The problem comes from cutting off your x integral at C. If the demand x is greater than C, then you still make a profit. The function you really want the expectation value of is min(C,x). Integrate that from 0 to 40.

 

[e(ho0n3]

Setting the upper limit of the integral to C implies that the demand can be at most C. The demand can be greater than C as Dick pointed out. The upper limit should be 40 since anything above 40 does not contribute to value of the integral.

But why min(C,x) though? The expected value of the demand should not be influenced by C.

 

[Dick]

If the demand is greater than C, then you still can only sell C. You don't want the expectation value of the demand, you want the expectation value of how much you can sell.

 

[1daj]

confused

The problem comes from cutting off your x integral at C. If the demand x is greater than C, then you still make a profit. The function you really want the expectation value of is min(C,x). Integrate that from 0 to 40.

Function of min(C,x)? I'm confused as to how to construct this.

 

[Dick]

Function of min(C,x)? I'm confused as to how to construct this.

If x<C then it's x. If x>C then it's the constant C. Just split your integral in two, between 0 and C it's what you have. From x in the range of C to 40, it's just the constant C.

 

[e(ho0n3]

If the demand is greater than C, then you still can only sell C. You don't want the expectation value of the demand, you want the expectation value of how much you can sell.

Ah, OK. Makes sense.

 

[1daj]

If x<C then it's x. If x>C then it's the constant C. Just split your integral in two, between 0 and C it's what you have. From x in the range of C to 40, it's just the constant C.

I've solved integral 0 to c (1/40)x dx + integral c to 40 c dx and got an answer that is scaring me. I am assuming I have mistaken what you meant. What is the reason that the second integral is set up the way it is? Sorry, about this, I am really confused.

 

[Dick]

You mean integral from c to 40 of c/40, right? Expectation value is the integral of value*probability density. You have a 1/40 in the first one, you should have a 1/40 in the second one. The reason you are splitting it up is because the amount you can sell maxes out at c for x>c. As I said before, you don't want expectation of demand, you want expectation of what you can sell.