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I am asked to find the joint pdf of $f(x,y)=128e^{(-8x-8y)}$ where $0 < x < y< \infty$ or $0$ otherwise. find the following: $$E(X)= ?$$

$$E(Y)= ? $$

$$E[X(Y-X)]= ? $$

Žiga Sajovic
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Madison
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  • I have added MathJax to your question. This is stackExchange, and not "solve my homework". Therefor, show us what have you tried and we will help you with the problem you are having when solving the problem.

    Hint: $$E(X)=\int_{x}^{\infty}\int_{0}^{\infty}xf(x,y)dxdy$$, etc.

     – Žiga Sajovic Nov 29 '17 at 04:54

1 Answers1

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Here if you see, the marginal of X is a Exponential Distribution with mean 1/16.

Thus E(X) = 1/16

And for the marginal of Y it is coming like this,

f(y) = 16e^(-8y)(1 - e^(-8y)) ; y>0

= 0 ; otherwise.

Thus by integrating you can find E(Y) = 3/16.

And for E [ X( Y - X )] = E(XY) - E(X^2)

By calculating you can see E(XY) = 1/128

And E(X^2) = Var( X ) + [ E( X ) ]^2

Thus X ~ Exponential Distribution with mean 1/16

Var( X ) = 1/256 and [ E(X) ]^2 = 1/256

E ( X^2) = 1/128

Thus E[ X(Y - X )] = 1/128 - 1/128 = 0

That's your answer.

And since I don't know the use of latex I have write in this manner. And if any help regarding Statistics is required, you can message me. I hope it helps you.

Soumajit Das
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