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

In summary, the conversation is discussing the process of taking a derivative of (1-F(x/a))(x) using a CDF function, a parameter, and the chain rule. The question is whether the derivative should be -(1-F(x/a))f(1/a) or -(1-F(x/a))f(x/a), and how to interpret the maximum result of F(x/a)f(x/a)=1. The conversation also includes a discussion about modeling a scenario where a seller chooses a price based on a probability function of wealth. The average "income" or "payoff" for the seller is calculated to be A = x*(1-F(x/a)). The conversation concludes with a clarification that the derivative
  • #1
ruzbayhhi
13
0
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.
 
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  • #2
ruzbayhhi said:
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?
 
  • #3
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.
 
  • #4
ruzbayhhi said:
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.
 
  • #5
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)
 

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

1. What does the expression (1-F(x/a))(x) represent?

The expression (1-F(x/a))(x) represents the derivative of a function with respect to the variable x, where the function is multiplied by the difference between 1 and the value of the cumulative distribution function F evaluated at x/a.

2. How can the derivative of (1-F(x/a))(x) be interpreted?

The derivative of (1-F(x/a))(x) can be interpreted as the rate of change of the function with respect to x, taking into account the impact of the cumulative distribution function on the overall result.

3. What is the solution to the derivative of (1-F(x/a))(x)?

The solution to the derivative of (1-F(x/a))(x) is given by the product rule of differentiation, where the derivative of (1-F(x/a)) is multiplied by x and the derivative of x is multiplied by (1-F(x/a)).

4. What are some common applications of the derivative of (1-F(x/a))(x)?

The derivative of (1-F(x/a))(x) is commonly used in statistics and probability, where it can be used to calculate the expected value and variance of a random variable following a given distribution.

5. Are there any special cases of the derivative of (1-F(x/a))(x)?

Yes, there are a few special cases of the derivative of (1-F(x/a))(x) that are commonly used in statistics, such as the derivative of the normal distribution (1-Φ(x))x, the derivative of the lognormal distribution (1-Φ(lnx))x, and the derivative of the exponential distribution (1-e^(-x))x.

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