This is a question from a previous exam. It seems too easy for the high amount of points it was worth. I don't have the answer key. Critique my answers please?

$X$ and $Y$ are independent uniform random variables with ranges (0,1) and (0,1). Define $W$ as the maximum of ($X$, $Y$): $W$=Max($X$,$Y$).

  1. Compute $P(W\leq0.3)$

$P(W\leq0.3) = P(X\leq0.3)\cdot P(Y\leq0.3) = 0.09$

  1. Compute $P(W=X)$

$0.5$ by symmetry

  1. Compute $E[W]$

$\int_0^12w^2dw = \frac{2}{3}$

  1. Compute $E[W|W=X]$

$\frac{2}{3}$ by symmetry and independence. I interpreted $W=X$ to mean the realized value of W is equal to the realized value of X. Is that the correct interpretation?

  • 1
    $\begingroup$ You can get nicer spacing for conditional probabilities by using \mid instead of |. $\endgroup$
    – joriki
    Aug 11 '18 at 17:13

Your answers are all right, though your reasoning is a bit terse. For $E[W]$, it's not clear to me what you computed – I guess you took the cumulative distribution function from $1.$, differentiated it and integrated it with $w$? I don't see where you need independence in $4.$; I think the result follows from $3.$ simply by symmetry. Here and in $2.$, to be precise you need not only symmetry but also $P(X=Y)=0$. I think your interpretation of the notation $E[W\mid W=X]$ is correct.


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