# Joint distribution

• Oct 4th 2008, 07:55 PM
kman320
Joint distribution
I am given X has distribution f(x)= (1/σ)*e^(-x/σ) and Y has the same distribution but with μ and y in place of sigma and x. I am also given Z=min{X,Y} and W={1 if Z=X and 0 if Z=Y}.
I am told to find the joint distribution of Z and W and also prove Z and W are independent.
I am stuck on this one. Does it break down into two cases (Z=X and Z=Y)?
• Oct 5th 2008, 06:25 AM
awkward
Quote:

Originally Posted by kman320
I am given X has distribution f(x)= (1/σ)*e^(-x/σ) and Y has the same distribution but with μ and y in place of sigma and x. I am also given Z=min{X,Y} and W={1 if Z=X and 0 if Z=Y}.
I am told to find the joint distribution of Z and W and also prove Z and W are independent.
I am stuck on this one. Does it break down into two cases (Z=X and Z=Y)?

kman320,

Are you sure of your problem statement? It looks to me like Z and W are dependent.
• Oct 5th 2008, 10:35 AM
kman320
Its seems they are, but he gives a hint. Show the prob. that Z<z given W=i (for i=1,2) is equal to the prob. Z<z.
Show P(Z<z l W=i)=P(Z<z) .
How will I go about doing the first step of finding the joint dist. of Z & W?
• Oct 7th 2008, 08:07 AM
kman320
Anyone?