Calculation of E[X|X>Y] for Exponential Random Variables

[mertcan]

Hi, Initially X and Y are exponential random variables with rate respectively $$\mu \lambda$$, and I am aware that E[X|X>Y] is obtained using joint distribution but I can not build up the integral structure, I intuitively think the result is just 1/mu, but I can not prove it to myself could you help me about that and building the integral structure?

 

[andrewkirk]

Start by writing the double integral that calculates the unconditional expectation $E[X]$. Use $x$ as integration variable for the outer integral, and $y$ for the inner integral. Once you've done that, only a minor adjustment is needed to the inner integration limits to turn it into the conditional expectation $E]X|X>Y]$.

 

[mertcan]

hi, I tried to do my work related to E(X1|X1<X2)*P(X1<X2), and X1, X2 are exponential random variables with rate respectively $$\lambda, \mu$$ I found a answer but I think it is wrong so could you tell me which part of my work is wrong?? ( I also looking forward to your answers @andrewkirk @Ray Vickson :) )

 

[andrewkirk]

Did you read my post? I told you that you need a double integral. Why have you tried to do something using only a single integral? The image you posted is too dark and smudgy to make out in detail what it says but even at a glance one can see that it only has single integrals, not double integrals.

 

[mertcan]

@andrewkirk @Ray Vickson I upload my work 2 ,let me express again that E(X1|X1<X2)*P(X1<X2), and X1, X2 are exponential random variables with rate respectively $$\lambda_1,\lambda_2$$I found a answer but I think it is wrong so could you tell me which part of my work is wrong?? ( by the way I did my best to make it not dark, when I upload, the top and bottom parts get dark a little )

 

[EngWiPy]

Hi, Initially X and Y are exponential random variables with rate respectively $$\mu \lambda$$, and I am aware that E[X|X>Y] is obtained using joint distribution but I can not build up the integral structure, I intuitively think the result is just 1/mu, but I can not prove it to myself could you help me about that and building the integral structure?

Are X and Y independent? In any case, you first need to find $$\mathbb{E}[X|Y, X>Y]$$. To do this, first write $$\mathbb{E}[X]$$, and then change the lower limit. After finding $$\mathbb{E}[X|Y, X>Y]$$, you will need to average over all values of Y. If you go through these steps, you should be able to find what you want.