- #1

- 98

- 0

## Homework Statement

Let X

_{1}, X

_{2}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 {X

_{1}, X

_{2}}

and Z = max {X

_{1}, X

_{2}}

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:

F

_{Y,Z}(y,z) = P( Y < y, Z < z)

= P(Y,Z<y) + P(Y<y, y<Z<z)

= P(X

_{1}<y, X

_{2}<y) + P(X

_{1}<y, y<X

_{2}<z) + P(X

_{2}<y, y<X

_{1}<z)

I really would like to know if this line is valid.. from here I used independence of Y,Z to find that

F

_{Y,Z}(y,z) = 1 - exp(-y(λ+μ)) - exp(-λz) - exp(-μz) + exp(-μy-λz) + exp(-λy-μz)

for y<= z.

so

f

_{Y,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!