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setrajonas
Registered User regular

Tomorrow's my Probability/Statistics final, and I was just going over some review questions, wherein I got stonewalled by this one:

Let X1 and X2 be independently and identically distributed with a common probability density function f(x)=(lamba)[e^(-lamba * x)] for x>0 and lamba>0. Let U = X1 + 2 * X2 and V = 2 * X1 + X2. What is the joint probability density function g(u,v) for u>0 and v>0?

I have no clue how to proceed from here, seeing as U and V look exactly identical.

Let X1 and X2 be independently and identically distributed with a common probability density function f(x)=(lamba)[e^(-lamba * x)] for x>0 and lamba>0. Let U = X1 + 2 * X2 and V = 2 * X1 + X2. What is the joint probability density function g(u,v) for u>0 and v>0?

I have no clue how to proceed from here, seeing as U and V look exactly identical.

0

## Posts

So you would do 2 * X2 and then add X1 for U.

And you would do 2 * X1 and then add X2 for V.

I have to think about this a little more as it's been a while since I did stats, but I thought I'd mention the above.

I should know this. Ruh oh.

m(theta) = E[e^(theta*x)] = Expectation of (e^(theta*x))

In this case, the mgf is

m(theta) = E[e^(theta*(x1 + 2x2))]

= E[e^(theta*x1) * e^(theta*x2) * e^(theta*x2)]

= E[e^(theta*x1)]*E[e^(theta*x2)]*E[e^(theta*x2)] (since x1 and x2 are iid)

= (lambda/(lambda-theta))^3

= mgf of erlang pdf where n=3

Use a similar process for part 2 of your question.

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I could follow that all the way up to "= E[e^(theta*x1)]*E[e^(theta*x2)]*E[e^(theta*x2)] (since x1 and x2 are iid)", then I'm not sure how you got to the next step. Also, we didn't learn the Erlang pdf in this course, so I somehow doubt that it's part of the answer, heh.