Die Question with Random Variables

Question

I have a homework question:

Roll a fair die, and let d be contained in $\{1,2,3,4,5,6\}$ . Then sample $d$ independent uniform random variables on $[0,1]$ and let $Y$ be the maximal of these random variables.

Find: $P(Y \leq y)$ and $E(Y)$.

Answer 1

Take a sample of $k$ independent uniforms, and let $W_k$ be the maximum of these. Then the probability that $W_k$ is $\le w$ is the probability that all the uniforms are $\le w$. This is $w^k$.

Now we are using a sample of $1$ with probability $\frac{1}{6}$, a sample of $2$ with probability $\frac{1}{6}$, and so on up to a sample of $6$ with probability $\frac{1}{6}$. It follows that $$\Pr(Y\le y)=\frac{1}{6}\left( y+y^2+y^3+\cdots +y^6 \right)$$ (for $0\lt y\lt 1$).

For $E(Y)$, there are many possible approaches. We can use the above expression for the cumulative distribution function, differentiate to find the density function, and then find $E(Y)$ as usual. Or if you know a formula for the expectation in terms of the cdf, you can use that. Or we can find the expectation of the $W_i$, for $1\le i\le 6$, and use a conditional expectation argument.