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

Question

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?

Answer 1

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.