(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[1] A random variable X is distributed as f_{X}(x) = 1/9*(1+x)^21{-1<= x<= 2}.

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

b) Find the cummulative distribution function of Y = X1{-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) = 31{0<=y<=1/4} + 1/31{1/4<y<1}.

3. 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.

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# Homework Help: Two problems with random variables transformations

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