- #1
Zoe-b
- 98
- 0
Homework Statement
Let X1, X2 be exponential RVs with parameter λ, μ respectively. (The question does NOT say that they are independent but I think this must be a typo?! If not I have even less idea how to do the question).
Let Y= min {X1, X2}
and Z = max {X1, X2}
Let W = Z-Y
Calculate the joint pdfs of (Y,Z), (Z,W), (Y,W). Which pairs are independent.
Homework Equations
I know how to find the the pdf of Y, Z seperately (via the cdf) but this doesn't seem to be directly relevant. Clearly Y,Z are not independent, so I think I need to find their joint pdf by first finding the joint cdf and then integrating.
The Attempt at a Solution
So far I have:
FY,Z(y,z) = P( Y < y, Z < z)
= P(Y,Z<y) + P(Y<y, y<Z<z)
= P(X1<y, X2<y) + P(X1 <y, y<X2<z) + P(X2<y, y<X1<z)
I really would like to know if this line is valid.. from here I used independence of Y,Z to find that
FY,Z(y,z) = 1 - exp(-y(λ+μ)) - exp(-λz) - exp(-μz) + exp(-μy-λz) + exp(-λy-μz)
for y<= z.
so
fY,Z(y,z) = 0 for y<z
= λμ(exp(-μy-λz) + exp(-λy-μz)) otherwise
This looks promising on the basis that the integral first wrt y from 0 to z, and then wrt z from 0 to infinity is 1.
But if I attempt to find the marginal distribution by integrating from 0 to ∞ wrt z, I get:
μexp(-μy) + λexp(-λy)
which is not what I would expect- I think the answer should be
(μ+λ)exp(-(μ+λ)y)
Am I even on the right lines at all? Its a much longer question but if this is wrong (as I fear it might be) there's not much point trying to plow through the rest using the wrong formulas..
Thank you!