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## Homework Statement

[1] A random variable X is distributed as f

_{X}(x) = 1/9*(1+x)^2

**1**{-1<= x<= 2}.

a) Find the density function of Y = -X^2 + X + 2.

b) Find the cummulative distribution function of Y = X

**1**{-1<=X<=1} +

**1**{X>=1}

[2] Find the function that transforms a variable X with f

_{X}(x) = e^(-x)

**1**{x>0} into a variable Y with f

_{Y}(y) = 3

**1**{0<=y<=1/4} + 1/3

**1**{1/4<y<1}.

## The Attempt at a Solution

The problem with exercise 1 is that the functions aren't injective in the domain of X. My first guess with a) was to use equivalent events:

F

_{Y}(y) =

**P**(Y<=y) =

**P**(-X^2 + X + 2 <= y)

F

_{Y}(y) =

**P**(-X^2 + X + 2 - y<=0) =

**P**([tex]X \le \frac{1}{2} - \frac{{\sqrt {9 - 4y} }}{2}[/tex]) +

**P**([tex]X \ge \frac{1}{2} + \frac{{\sqrt {9 - 4y} }}{2}[/tex])

Then, taking into account that X is absolutely continuous:

F

_{Y}(y) =

**P**([tex]X \le \frac{1}{2} - \frac{{\sqrt {9 - 4y} }}{2}[/tex]) + (1 -

**P**([tex]X \le \frac{1}{2} + \frac{{\sqrt {9 - 4y} }}{2}[/tex]))

F

_{Y}(y) = F

_{X}([tex]\frac{1}{2} - \frac{{\sqrt {9 - 4y} }}{2}[/tex]) + (1 - F

_{X}([tex]\frac{1}{2} + \frac{{\sqrt {9 - 4y} }}{2}[/tex])).

Then I could derive term to term and find f

_{Y}(y). Is this OK? I'm mainly worried about the unicity of all this, due to g(x) not being injective.

For point b), on the other hand, I don't know how to interprete it. Should I consider this a problem of conditional probabilities? I mean, F

_{Y}(y) = F

_{X|-1<=X<=1}(x) + ????

For exercise 2, I need a hint. Best thing I could think was that, since f

_{Y}(y) = f

_{X}(g

^{-1}(y))/(g'(g

^{-1}(y))), that in both cases g(x) must some sort of logarithmic function.

Any recomendations? Thanks.