# Solving expected value problem with logistic function

by tlonist
Tags: expected value, logistic model, probability, solvability
 P: 1 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?

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