- #1
Dassinia
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Hello, want to know if it's correct
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]
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
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