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Mentallic

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## Homework Statement

If we have the exponential distribution [tex]f_X(x)=\frac{1}{2}e^{-x/2}[/tex] then show that the cumulative distribution function of [tex]Y=\sqrt{X}[/tex] is given by [tex]F_Y(y)=1-e^{-y^2/2}[/tex]

## Homework Equations

[tex]F_Y(y)=f_X(x)\cdot\left| \frac{dx}{dy}\right|[/tex]

[tex]F_Y(y)=f_X(h^{-1}(y))\cdot\left| \frac{d(h^{-1}(y))}{dy}\right|[/tex]

## The Attempt at a Solution

[tex]Y=\sqrt{X}=h(x)[/tex]

[tex]\therefore h^{-1}(y)=y^2[/tex]

[tex]\frac{d(h^{-1}(y))}{dy}=2y[/tex]

[tex]f_X(h^{-1}(y))=\frac{1}{2}e^{-y^2/2}[/tex]

[itex]\therefore[/itex] after plugging these values into the formula in the relevant equations,

[tex]F_Y(y)=y\cdot e^{-y^2/2}[/tex]

Which is not what I was meant to show. I only had one example in my text book to go off of and I (from what I can tell) think I applied it correctly to my question, but clearly I haven't. Can someone please guide me in the right direction, and also if you can see anything in my steps that need to be scrutinized, don't be afraid to speak out.