I am looking for several options of modeling the probability of people agreeing to do some work depending on the price/payment. The payment can only range between $p_1$ and $p_2$, $(p_1 < p_2)$. I looked at normal distribution and not sure if I can use it to model my case because the higher the payment, the higher probability that people will do it, intuitively, therefore it's monotonic. Is there any way to only use the left side of the normal distribution to preserve the monotonicity?
Any other suggestions would be appreciated. Thanks.
If you want a normal-type distribution, you could consider a folded normal distribution. Essentially, if $X$ is a normally distributed random variable, then the random variable $Y=|X|$ follows a folded normal distribution. In particular, if $X$ has mean $0$, then $Y=|X|$ follows a half normal distribution, which decreases monotonically. Thus $1-Y$ increases monotonically as you tend to infinity. If you wanted a bounded interval, you could cut the function off at some point and then scale as needed.