ACJC Discrete Random Variables Tutorial Q1
A biased six-sided die is such that $\text{P}\left( X=1 \right)=p$ and $\text{P}\left( X=r+1 \right)=\frac{1}{2}\text{P}\left( X=r \right)$, $r=1$, $2$, $3$, $4$, $5$ where $X$ denotes the number on the uppermost face of the die when it is thrown on the table.
Show that $p=\frac{32}{63}$ and find the value of $\text{E}\left( X \right)$.
This biased die and an ordinary fair six-sided die are thrown at the same time. If $Y$ denotes the number on the uppermost face of the ordinary die, write down the expectation of $Y$.
Find also
(a)
the value of $\text{P}\left( X+Y=4 \right)$, and
(b)
$\text{E}\left( X+Y \right)$.
Deduce the value of $\text{E}\left( S \right)$, where $S$ denotes the total score of the numbers on the other ten faces of the two dice.
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