Consider the random experiment in which three fair dice are rolled simultaneously (and independently).
Let $X$ be the random variable defined as the sum of the values of these three dice.
Let $Y_1$ be the maximum of the three values, let $Y_2$ be their product, and let $Y=Y_1+Y_2$.
Finally, define $Z=E[X∣Y]$ (the conditional expectation of $X$ given $Y$).
Find the expected value of the random variable $Z$. Enter your answer as a fraction, such as $\frac{3}{2}$.
I would like some help on where to start this problem, please. I know $X$ can be any value between $3$ and $18$ and $Y_2$ could be (most of the values) between $1$ and $216$; i.e., $Y_2$ will never be $7$, $11$, $13$, etc. since those numbers are prime and are greater than $6$.
Thanks!
Best Answer
The expected value of a die roll is 3.5. Thus, for three independent rolls, the expected value is 10.5. $\frac{21}{2}$, if you're being picky.
Conditional expectation has the following property, known as averaging: $E( E(X|Y) ) = EX$.