Applied Stochastic Processes - 2?

[ra_forever8]

Two independent random variables X and Y has the same uniform distributions in the range [-1..1]. Find the distribution function of Z=X-Y, its mean and variance.

=Using change of variables technique seems to be easiest.

fX(x) = 1/2

fY(y) =1/2

f = 1/4 ( -1<X<1 , -1<Y<1)

Using u =x -y , v= x+y

Jacobian is del (x,y) / del (u,v) = 1/2

then J =1/2

and g (u,v) =1/8

Integrate g with respect to v then

gu = (u+2)/4 -2<u<0

and gu = ( -u+2) /4 , 0< u<2

is the PDF of u

Finally mean and variance, Can someone help me? Thanks

 

[Ray Vickson]

Two independent random variables X and Y has the same uniform distributions in the range [-1..1]. Find the distribution function of Z=X-Y, its mean and variance.

=Using change of variables technique seems to be easiest.

fX(x) = 1/2

fY(y) =1/2

f = 1/4 ( -1<X<1 , -1<Y<1)

Using u =x -y , v= x+y

Jacobian is del (x,y) / del (u,v) = 1/2

then J =1/2

and g (u,v) =1/8

Integrate g with respect to v then

gu = (u+2)/4 -2<u<0

and gu = ( -u+2) /4 , 0< u<2

is the PDF of u

Finally mean and variance, Can someone help me? Thanks

You don't need to know g(u) to do these last two questions. However, since you have already obtained g(u), what is stopping you from using it to compute EZ and Var(Z) directly?

Also: please change the title of new posts on similar topics; we try to use titles to keep thing straight, and confusion can result when more than one of post from the same person has the same title. You could even use the same title but with suffixes such as 1,2, etc, or a,b,c,... .

Mod note: Changed thread title...

 

[Mark44]

Questions involving integration and Jacobians are more suited to the Calculus & Beyond section. I am moving the thread to that section.