Need a formula for a one of these gamma densities

#1
For the integral from [0, infinity] for x^(s-1) * e^(-lambda * x)

You have s and lambda, so is the final answer for this (s-1)! / lambda^(s) ? Or is it (s-1)! / lambda ^(s-1) ?

I understand this, but the only question I have is in the denominator.
 

Dason

Ambassador to the humans
#2
Are you assuming s is an integer? Otherwise you should probably use the gamma function. Anywho, can you think of anyway to check which of those answers is correct? Possibly using the fact that a gamma density integrates to 1?
 
#3
Are you assuming s is an integer? Otherwise you should probably use the gamma function. Anywho, can you think of anyway to check which of those answers is correct? Possibly using the fact that a gamma density integrates to 1?
Yes, s is the integer, and this gamma density integrates to 1.

My question is for the lambda part in the denominator, is it lambda^(s) or lambda^(s-1) ?
 

Dason

Ambassador to the humans
#4
The main part of the post is saying you should be able to deduce which it is by using the fact that a gamma density integrates to 1. Which of the two do you think it is and why?
 
#5
The main part of the post is saying you should be able to deduce which it is by using the fact that a gamma density integrates to 1. Which of the two do you think it is and why?
It's the one with lambda ^ s, just remembered the entire gamma density formula, wrote it out, and this one is the one for the gamma density to integrate to one.