# Solving expected value problem with logistic function

• tlonist
In summary, the conversation discusses an expected value problem involving a desired value and a variable amount. The probability of success is determined by a logistic function and results in a reward or loss. The speaker is attempting to solve for the correct value to reach the desired expected value, but has not been able to solve it by hand or using Matlab. They inquire about simplifying the solution and a potential solution is suggested.
tlonist
I have an expected value problem where z is a desired expected value and I want to reach and x is an amount I can vary.

There is a probabilty of success based on a logistic function ρ(x) with a reward of λx and failure with a probability of (1-ρ(x)) and loss of x. I am trying to solve for the correct value of x to reach an expected value z.

So:

$$z = p(x) \lambda x - (1-p(x)) x$$

$$z = \frac{\lambda x}{1+ e^{-a-bx} } + \frac{x}{1+ e^{-a-bx} } -x$$

I tried solving in Matlab but it says there is no explicit solution and I haven't been able to solve by hand.

What would be the next course of action to solve this? Is there a way to simplify?

It looks as if ##p(x)=\dfrac{1}{1+e^{-a-bx}}## is a solution.

## 1. What is the expected value problem with logistic function?

The expected value problem with logistic function is a mathematical concept used to determine the average outcome or value of a random variable that follows a logistic distribution. This problem is often encountered in statistics, economics, and machine learning.

## 2. How do you solve an expected value problem with logistic function?

To solve an expected value problem with logistic function, you need to use the formula E(x) = a + b / (1 + c * e^(-kx)), where a, b, c, and k are constants. First, determine the values of these constants based on the given data. Then, plug them into the formula to calculate the expected value.

## 3. What is the significance of solving an expected value problem with logistic function?

Solving an expected value problem with logistic function allows us to make predictions and decisions based on the average outcome of a random variable. This is useful in various fields such as finance, marketing, and supply chain management, where understanding the expected value can help in making informed decisions.

## 4. Can the logistic function handle both positive and negative values?

Yes, the logistic function can handle both positive and negative values. The shape of the logistic curve is symmetrical, which means it can model both increasing and decreasing trends. Therefore, it can be used to solve expected value problems for a wide range of data.

## 5. Are there any limitations to solving expected value problems with logistic function?

One limitation of using the logistic function to solve expected value problems is that it assumes a continuous and smooth distribution of data. This may not always be the case in real-world scenarios, where the data may have outliers or gaps. Additionally, the logistic function may not be suitable for solving expected value problems for data that follows a non-logistic distribution.

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