Polynomial transformation of random variable

[raynard]

Homework Statement

Given a random variable X with a known distribution (e.g. a beta distribution), find the distribution of
f(X) = X^2 + X

The Attempt at a Solution

I've tried the normal approaches: the standard transformation theorem; conditioning on X; Laplace transformation, etc. They don't seem to work. Any hints?

 

[Tomer]

Here's a hint:
http://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables

Look under "Functions of random variables".
There's a formula there. Notice that your function isn't invertible on the whole line but you can easily solve this by separating your domains...

 

[raynard]

Thanks for the reply!

I tried the standard method you described, but I see no easy way to find the inverse of

f: x -> x^2 + x

 

[Tomer]

Yup, I don't think it's very pretty...
I have to say, I'm pretty rusty when it comes to probability. I remember the best ways to solve this type of questions is to write Y = X² + X, and then:
P(Y < y) = P (X² + X < y) = P(X² + X - y < 0 ) = ...

Then you'd have to solve this inequality and proceed from there...
But maybe there are shortcuts. I just responded cause I saw no one else did :-)