Derivative of (1-F(x/a))(x): Interpretation & Solution

[ruzbayhhi]

I am trying to take a derivative of (1-F(x/a))(x) (where F(.) is a CDF and a is a parameter), and I an not sure whether the derivative should be :
-(1-F(x/a))f(1/a)
or
-(1-F(x/a))f(x/a).

Also, I am not sure how to interpret the result that at the maximum:
F(x/a)f(x/a)=1.

 

[torquil]

I am trying to take a derivative of (1-F(x/a))(x) (where F(.) is a CDF and a is a parameter).

This expression seems a bit strange. The part within the first parenthesis, 1-F(x/a), is just a real number, but you seem to be evaluating it at x?

 

[ruzbayhhi]

I am modelling a scenario where a seller chooses a price, x. If the buyer has less than a threshold amount of money (w <= x/a) he doesn't buy the product (payoff for seller = 0). If he has enough money, he will buy it (payoff = x). The seller doesn't know exactly how much money the buyer has and had to decide based off a probability function of wealth.

 

[torquil]

I am modelling a scenario where a seller chooses a price, x. If the buyer has less than a threshold amount of money (w <= x/a) he doesn't buy the product (payoff for seller = 0). If he has enough money, he will buy it (payoff = x). The seller doesn't know exactly how much money the buyer has and had to decide based off a probability function of wealth.

From this description, I calculated A, the average "income" or "payoff" for the seller, to be the same result as you:

A = x*(1-F(x/a))

I don't want to risk doing other peoples assignments and so on, but I will say that the chain rule of differentiation applied to F(x/a) would be

d/dx F(x/a) = f(x/a) * 1/a

where f is the probability density corresponding to the CDF F. Reason: you differentiate F(x/a) with respect to ITS argument x/a, and get f(x/a). Then you multiply this by the derivative of THAT argument x/a with respect to x, which is 1/a.

 

[HallsofIvy]

Since there are no differential equations involved here, I am moving this thread.

What is "f"? The derivative of F?

If so then the derivative of (1- F(x/a))x is
(1- F(x/a))- f(x/a)(x/a)