An unbiased disc has a single dot marked on one side and two dots marked on the other side. The disc

Question

An unbiased disc has a single dot marked on one side and two dots marked on the other side. The disc and an unbiased die are thrown and the random variable $X$ is the sum of the numbers of dots showing on the disc and on the top of the die. Tabulate the probability distribution of $X$. Write down $\mathrm{E}(X)$ and hence show that $\mathrm{Var}(X) = \frac{19}{6}$. Find $\mathrm{P}(X \geq 4 | X \leq 7)$. Two independent observations of $X_1$ and $X_2$ are taken of $X$. Find $\mathrm{Var}(2X_1 - X_2 + \mathrm{Var}(2X_1 - X_2))$. Find $\mathrm{P}(X_1 - X_2 \geq 5)$. Hence find $\mathrm{P}(|X_1 - X_2| \geq 5)$.

Accepted Answer

$\mathrm{P}(X\ge4\mid X\le7)=\frac{8}{11}$; (a) $\frac{95}{6}$ (b) $\frac{5}{144}$, hence $\frac{5}{72}$

N1995 P2 Q6 [Modified]

N1995 P2 Q6 [Modified]

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