Function of random variable, limits of integration

[Gauss M.D.]

Homework Statement

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

Homework Equations

The Attempt at a Solution

F_Y(Y) = P(Y < y) = P(2X² + 1 < y) = P(X < +\sqrt{1/2(y-1)} = F_X(+\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 F_Y(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!

 

[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?

 

[HallsofIvy]

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?

 

[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...