- #1

diracdelta

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## Homework Statement

Random variable X is uniformly distributed on interval [0,1]:

[tex]f(x)=\begin{cases} 1 & \text{ if } 0\leq x\leq 1\\ 0 & \text{ else} \end{cases}[/tex]

a) Find probability density function ρ(y) of random variable [tex]Y=\sqrt{X} +1[/tex]

I tried like this. Is it good, if no why not?

[tex]F_{Y}(y)=P(Y\leq y)=P(x^{2}+1\leq y)=P(x\leq y^{2}+1)=F_{x}(y^{2}+1) \\ f_{X}(x)=F_{X}'(y^{2}+1)=f_{X}(y^{2}+1)* \left | \frac{d}{dy} (y^{2}+1)\right |=f_{X}(y^{2}+1)*2y \\ f_{Y}(y)=\begin{cases} 2y & \text{ if } 1\leq x\leq 2\\ 0 & \text{ else} \end{cases}[/tex]