# Conditional Probability Density Function of a single Random Variable With A Gamma Distribution

I am trying to find the probability density function of a Gamma distribution given that x > 4. I thought that I would be able to take the density and simply set it back 4 so that the domain would be appropriate meaning:

Instead of $\frac{\beta^\alpha}{\Gamma(\alpha)} * x^{\alpha -1}*e^{-\beta * x}$

I do

$\frac{\beta^\alpha}{\Gamma(\alpha)} * (x -4 )^{\alpha -1}*e^{-\beta * (x - 4)}$

I take that density and divided by the conditional probability of x > 4 to get the conditional density.

However this is incorrect.

1.) Why is dividing the density by the conditional probability the incorrect approach here.

2.) What is the correct approach?

The correct answer is $\frac {f(x)} {P\{X>4\}}$ for $x>4$ and 0 for $x<4$ where $f$ is the given Gamma density. To arrive at this consider $\frac {P\{X \leq x,X>4\}} {P\{X>4\}}$. This is 0 if $x<4$. For $x >4$ we get $\frac {\int_4^{x} f(y)dy} {P\{X>4\}}$. Differentiating this we get the density as $\frac {f(x)} {P\{X>4\}}$. As far as 1) is concerned, dividing by $P\{X>4\}$ is right but shifting the density is not.