2022 DHS P2 Q11
Emma has a computer program that generates a random positive integer $X$. The probability distribution of $X$ is:
$\text{P}\left( X=r \right)=\frac{a}{{{r}^{3}}}$, $r\in {{\mathbb{Z}}^{+}}$ and $a$ is positive constant.
For the rest of the question, you may use the following results:
$\sum\limits_{m=1}^{\infty }{\frac{1}{m}}$ does not exist, $\sum\limits_{m=1}^{\infty }{\frac{1}{{{m}^{2}}}=\text{1}\text{.6449}}$ and $\sum\limits_{m=1}^{\infty }{\frac{1}{{{m}^{3}}}=\text{1}\text{.2021}}$.
(a)
Find the value of $a$.
[2]
(b)
Find $\text{E}\left( X \right)$ and explain why $\text{Var}\left( X \right)$ cannot be calculated.
[2]
(c)
Find $\text{P}\left( \left. X\ge 2 \right|X\le 15 \right)$.
[3]
Emma generates $10$ numbers using her program namely, ${{X}_{1}}$, ${{X}_{2}}$, … ,${{X}_{10}}$. The random variable $Y$ denotes the number of times the number ‘$3$’ occurs among the $10$ numbers. You can assume that the numbers generated by the program are independent of each other.
(d)
Find $\text{P}\left( Y>2 \right)$.
[2]
(e)
Find $\text{P}\left( {{X}_{1}}+Y=3 \right)$.
[4]
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