How to fit a cumulative distribution over two proportions at different values I am trying to figure how how I would fit a distribution to the data described below.
Imagine I run an experiment where in condition 1 subjects choose between option A costing \$50 and option B costing \$100. In condition 2 subjects chose between option A costing \$50 dollars and option B costing \$80.
So imagine that I get 20% preferring option B in condition 1, and 40% preferring option B in condition 2. To me that says that 20% of the population values B over A by at least \$50 and 40% of the population values option B over A by at least \$30.
Let's say I assume a normal distribution around how much people value B over A. It seems like if I know that 20% value the difference over \$50 and 40% value it over \$30 I should be able estimate the total distribution of this difference in the population.
I was also thinking that maybe I need one more condition to properly estimate. Is this correct? If so, what would be the right tool to estimate that distribution?
Thank you!
 A: Normality is a pretty strong assumption.
let $v$ be something that converts an event ("you get A") to a value in dollars, but that $v$ is a random variable indexed by the person whose preferences it is applied to. 
$P(v(B)-v(A)>30) = 0.4$
$P(v(B)-v(A)>50) = 0.2$
For simplicity let $X = v(B)-v(A) \sim N(\mu_{X},\sigma^2_{X})$
$P(X>30) = 0.4$  
$P(\frac{X-\mu_X}{\sigma_X}>\frac{30-\mu_X}{\sigma_X}) = 0.4$  
$\mu_X+\Phi^{-1}(1-0.4)\,{\sigma_X}=30$  .... (1)
similarly,
$\mu_X+\Phi^{-1}(1-0.2)\,{\sigma_X}=50$  .... (2)
check normal tables to find the $\Phi^{-1}$ values and solve the pair of simultaneous equations for the parameters.
If the proportions are exact and the assumptions are correct, this is sufficient. If the assumption might be wrong, you could use additional point(s) to assess the suitability of the approximation. If the proportions are estimates, you could estimate the standard errors in the parameter estimates from the uncertainties relating to the standard errors of proportions (and their correlation if need be).
