Consider the random experiment in which 3 fair dice are rolled simultaneously (and also independently). Let \$X\$ be the random variable identified as the sum of the worths of these 3 dice. Let \$Y_1\$ be the maximum of the 3 worths, let \$Y_2\$ be their product, and also let \$Y=Y_1+Y_2\$. Finally, define \$Z=E\$ (the conditional expectation of \$X\$ given \$Y\$).

Find the meant value of the random variable \$Z\$. Enter your answer as a portion, such as \$frac32\$.

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I would prefer some aid on where to start this trouble, please. I recognize \$X\$ deserve to be any type of value in between \$3\$ and \$18\$ and \$Y_2\$ could be (a lot of of the values) in between \$1\$ and also \$216\$; i.e., \$Y_2\$ will never be \$7\$, \$11\$, \$13\$, and so on since those numbers are prime and are higher than \$6\$.

Thanks!

The expected worth of a die roll is 3.5. Hence, for three independent rolls, the intended worth is 10.5. \$frac212\$, if you"re being picky.

Conditional expectation has actually the following home, recognized as averaging: \$E( E(X|Y) ) = EX\$.

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