# Max of 2 IID Uniforms and symmetry

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?

• You can get nicer spacing for conditional probabilities by using \mid instead of |. 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.