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**Hello, want to know if it's correct**

1. Homework Statement

1. Homework Statement

X and Y two random variables iid of common density f and f(x)=x*exp(-x²/2) if x≥0 and f(x)=0 if x≤0

and Z=min(X,Y)

Find

-The density of Z

-The density of Z²

- E[Z²]

## Homework Equations

## The Attempt at a Solution

1.[/B]

FZ(u) = P(min(X,Y )≤u)

= 1 − P(min(X,Y )>u)

= 1 − P(X>u)*P(Y>u)

= 1 − (1 − F(u))²

And F(u)=1-exp(-u²/2) if u>0 , 0 if u≤0

So FZ(u)=1-exp(-u²/2) if u>0 , 0 if u≤0

The density is then

fZ(u) = u*exp(-u²/2) 0 if u>0 , 0 if u≤0

2. I'm not sure here

u≤0

FZ²(u)=P(Z²≤u)=0

u≥0

Z²≤u , -√u≤Z≤√u

From -√u to 0 , FZ²(u)=0

From 0 to √u , FZ²(u)=FZ(u)=1-exp(-u²/2)

So Z² has the same density as Z ?

Thanks