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Expectation of a Maximum Using Conditional Expectation

Article Quant Q&A · Author: Eddie Ma
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Summary

The document asks whether the expected maximum of two discrete random variables can be decomposed by conditioning on which variable is larger, even when the variables are dependent. The accepted answer says the decomposition works without an independence assumption, but corrects the second conditional term: on the event that y exceeds x, the value contributing to the maximum is y, so that term must use the conditional expectation of y.

The justification is the law of total expectation applied to the two events that partition the outcomes: x is greater than y, or x is less than y. For a complete formulation when ties are possible, the event x equals y must also be included or assigned consistently to either region, since the stated partition leaves ties out. The source gives only a brief conceptual answer; it provides no detailed proof or worked numerical example.

Key ideas

  • The expected maximum can be split by conditioning on which variable is larger.
  • The variable inside each conditional expectation must be the variable that attains the maximum on that event.
  • The decomposition does not require x and y to be independent.
  • The law of total expectation supplies the justification.
  • Ties require an explicit event or a convention for assigning them.

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# Does E[max(x, y)] equal to E[x|x>y]*P(x>y) + E[x|x<y]* P(x<y) when x and y are not independent?


# Does E[max(x, y)] equal to E[x|x>y]*P(x>y) + E[x|x<y]* P(x<y) when x and y are not independent?












Suppose x and y are discrete random variable, I can write them in summation. And it seems like they are equal. Any ideas?

## Answer by George Dewhirst (score 0, accepted)

https://quant.stackexchange.com/a/45337

It's $E[y|x<y]* P(x<y)$ on the second expression but otherwise fine yes. We can prove it using the law of total expectation without independence of $x,y$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.