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## Homework Statement

Suppose a random variable X has probability density function(pdf)

f(x) { 1/3 for [tex]1 \leq x \leq 4[/tex]

find the density function of [tex] Y= \sqrt{X}[/tex]

## The Attempt at a Solution

[tex] y=g(x)=\sqrt{x}[/tex]

so [tex]g^-1(y)=x=y^2[/tex]

[tex]A= \{ x: 1 \leq x \leq 4 \}[/tex]

is monotonic onto

[tex]B= \{y: 1 \leq y \leq 2 \}[/tex]

[tex](g^-1(y))^'[/tex][tex]=2y[/tex]

[tex]f(y)=fx (g^-1(y). |(g^-1(y))^'[/tex][tex]|[/tex]

which gives me

[tex]f(x)=[/tex][tex] \{ 2y/3 [/tex] for [tex]1 \leq y \leq 2 \}[/tex]

This seems to be an invalid pdf as for y=2 f(y)=1.3333 which is >1

Can anyone tell me where I went wrong?

thanks