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

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In summary, you need a double integral to calculate the unconditional expectation of a random variable.
  • #1
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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?
 
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  • #2
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]##.
 
  • #3
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 :) )
 

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  • #4
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.
 
  • #5
@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 )
 

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  • #6
mertcan said:
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.
 

What is the meaning of "Expectation of E[X|X>Y]"?

The expectation of E[X|X>Y] refers to the expected value of a random variable X, given that X is greater than another random variable Y. In other words, it is the average value of X when it is known that X is larger than Y.

How is the expectation of E[X|X>Y] calculated?

The calculation of the expectation of E[X|X>Y] involves taking the sum of the products of each possible value of X and its corresponding probability, given that it is greater than Y. This can also be expressed as the integral of X multiplied by its conditional probability density function.

Why is the expectation of E[X|X>Y] important in statistics?

The expectation of E[X|X>Y] is an important concept in statistics because it allows us to understand the average value of a random variable X in a specific situation, where X is known to be larger than another random variable Y. This can provide valuable insights in various fields such as finance, economics, and engineering.

How does the expectation of E[X|X>Y] differ from the overall expectation of X?

The overall expectation of X takes into account all possible values of X, while the expectation of E[X|X>Y] only considers values of X that are greater than Y. This means that the overall expectation of X may be different from the expectation of E[X|X>Y] in situations where X is not always greater than Y.

Can the expectation of E[X|X>Y] be negative?

Yes, the expectation of E[X|X>Y] can be negative. This can occur if the values of X that are greater than Y are mostly negative, or if the probability of X being greater than Y is very small. In such cases, the average value of X given that it is greater than Y may be negative.

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