# Homework Help: Show the gamma density function integrates to 1

1. Jun 29, 2013

### Catchfire

1. The problem statement, all variables and given/known data
Show the gamma density function integrates to 1.

2. Relevant equations
Assume α > 0, λ > 0, t > 0
g(t) = $\frac{λ^α}{\Gamma (α)} t^{α-1}e^{-λt}$
$\Gamma (α)= \int_0^∞ t^{α-1} e^{-t} dt$
3. The attempt at a solution

Show $\int_0^∞ \frac{λ^α}{\Gamma (α)} t^{α-1}e^{-λt} dt = 1$

$\int_0^∞ \frac{λ^α}{\Gamma (α)} t^{α-1}e^{-λt} dt$
= $\frac{λ^α}{\Gamma (α)} \int_0^∞ t^{α-1}e^{-λt}dt$
= $\frac{λ^α}{\Gamma (α)} \int_0^∞ t^{α-1}e^{-t}e^λdt$
= $\frac{λ^αe^λ}{\Gamma (α)} \int_0^∞ t^{α-1}e^{-t}dt$
= $\frac{λ^αe^λ \Gamma (α)}{\Gamma (α)}$
= $λ^αe^λ$

...where did I lose the plot?

2. Jun 29, 2013

### Dick

You lost it when you said $e^{-λt}=e^{-t}e^λ$. That's not true. I would try the variable substitution $u=λt$.

Last edited: Jun 29, 2013
3. Jul 1, 2013

### Catchfire

Looks like I need to refresh myself on the laws of exponents.

That substitution did the trick, thanks.