# Bounding a truncated normal with a gamma

1. Feb 20, 2016

### FallenApple

So say I have a truncated normal. That is, N(mu,sigma) that is from 0 to infinity only.

I need to find a Gamma such that a constant C*Gamma(A,B) is always above N(mu, sigma). How would I go about finding such a A, B that would work given fixed mu and sigma?

2. Feb 21, 2016

### andrewkirk

Choose a value of $A$, the gamma's shape parameter. It will have to be a value that makes the gamma's pdf diverge to infinity at $x=0$, because otherwise the pdf will be zero there, and so will be below the normal pdf.
Let $f$ be the pdf of the gamma minus the pdf of the truncated normal. It will be a function of $x, A, B$, of which only two ($x, B$) are unknown.
Use numeric analysis to solve the following two equations for $x$ and $B$.

1. $f(x,A,B)=0$
2. $\frac{\partial f}{\partial x}(x,A,B)=0$.

Remove any solutions where $\frac{\partial^2 f}{\partial x^2}(x,A,B)\leq 0$.

What remains will be parameters of curves that have the property you seek. The gamma pdf curve will touch that of the normal, but not go below it. There may be more than one such curve.

You can repeat this for different admissible values of $A$ to get additional sets of solutions.