# Probability Distribution that fits my parameters?

I'm trying to create a PDF that has the max values at its tails, and a P(x) of 0 at its mean.

Essentially it would be something like two normal distributions lined up side to side.

Is there any literature regarding such a distribution? I'm trying to model the probability of exiting a position. If the price of the position doesn't move, it has the lowest probability of being exited, but the probability of exit is highest with an extreme move in either direction.

Essentially it would be something like two normal distributions lined up side to side.

This would e.g. be a mixture of two Gaussians. I.e. let $Y_+ \sim \mathcal{N} \left( \mu_+, \sigma_+^2 \right)$ and $Y_- \sim \mathcal{N} \left( \mu_-, \sigma_-^2 \right)$, then

\begin{equation} X \sim \begin{cases} Y_+ & \text{with probability } p\\ Y_- & \text{with probability } 1 - p \end{cases} \end{equation}

Then $X$ has the probability density function

\begin{equation} f_X(x) = p \phi_{Y_+}(x) + (1 - p) \phi_{Y_-}(x), \end{equation}

where $\phi_{Y_\pm}$ are the corresponding normal density functions.

• So in essence, I create two normal distributions and then essentially choose which distribution to sample from using a binomial probability function? – milkmotel Sep 6 '16 at 21:05
• Yes - exactly. These kinds of mixtures are quite common - e.g. in modelling two-sided jumps sizes. – LocalVolatility Sep 6 '16 at 22:28