# Function of random variable, limits of integration

1. Apr 28, 2013

### Gauss M.D.

1. The problem statement, all variables and given/known data

X is uniformly distributed over [-1,1]. Compute the density function f(y) of Y = 2X2 + 1.

2. Relevant equations

3. The attempt at a solution

FY(Y) = P(Y < y) = P(2X2 + 1 < y) = P(X < +$\sqrt{1/2(y-1)}$ = FX(+$\sqrt{1/2(y-1)}$)

We have that f(x) = 0.5 for -1 < x < 1, so we should integrate f(x) from -1 to +$\sqrt{1/2(y-1)}$ to get FY(y), and then take the derivative of that to get f(y).

But how do I deal with the + in front of the square root? What does P(X < +$\sqrt{1/2(y-1)}$ even mean? "The probability that X is smaller than plus minus x" doesn't seem to make much sense to me!!

2. Apr 28, 2013

### Gauss M.D.

I guess it means X is BETWEEN -$\sqrt{1/2(y-1)}$ and +$\sqrt{1/2(y-1)}$? But how do I figure out which bounds y has in that case?

3. Apr 28, 2013

### HallsofIvy

Staff Emeritus
You are told that X lies between -1 and 1. $2X^2+ 1$ is a parabola which has a minimum at Y= 1 (when X= 0) and a maximum of Y= 3 (at x= -1 and 1).

Now, the problem asks you to find the "probability density function" for Y so why are you integrating at all?

4. Apr 28, 2013

### Gauss M.D.

I'm trying to find F(y) so I can take the derivative and get f(y). Basically trying to follow lecture notes...