RVHS 2025 H2 Math Prelim Paper 2 — Discrete Random Variables (worked solution)

Discrete Random Variables: Distribution; E(X), Var(X) — RVHS 2025 H2 Math Prelim Paper 2

River Valley High School2025 PrelimPaper 2●○○ Introductory5 marks

What this question tests

Distribution; E(X), Var(X).

Question

In a game, a fair six-sided die is rolled. If a 1 or a 2 is rolled, then the die is rerolled and the score is the result of the reroll. Otherwise, the score is the number rolled originally.

Give the probability distribution function of the scores in the form of a table.
Show that the expected score is \(\dfrac{25}{6}\). Explain the meaning of this value in this context.
Find the exact variance of the scores.

Show full worked solution▾

(a) Let \(X\) be the r.v. denoting the score.

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)
\(\mathrm{P}(X=x)\)	\(\dfrac{1}{18}\)	\(\dfrac{1}{18}\)	\(\dfrac{2}{9}\)	\(\dfrac{2}{9}\)	\(\dfrac{2}{9}\)	\(\dfrac{2}{9}\)

\[\begin{aligned} \mathrm{P}(X=1) &= \left(\frac{1}{6}+\frac{1}{6}\right)\!\left(\frac{1}{6}\right) = \left(\frac{1}{3}\right)\!\left(\frac{1}{6}\right) = \frac{1}{18}\\[4pt] \mathrm{P}(X=2) &= \left(\frac{1}{6}+\frac{1}{6}\right)\!\left(\frac{1}{6}\right) = \frac{1}{18}\\[4pt] \mathrm{P}(X=3) &= \frac{1}{6}+\left(\frac{1}{3}\right)\!\left(\frac{1}{6}\right) = \frac{4}{18} \text{ or } \frac{2}{9}\\[4pt] \mathrm{P}(X=4) &= \frac{1}{6}+\left(\frac{1}{3}\right)\!\left(\frac{1}{6}\right) = \frac{2}{9}\\[4pt] \mathrm{P}(X=5) &= \frac{1}{6}+\left(\frac{1}{3}\right)\!\left(\frac{1}{6}\right) = \frac{2}{9}\\[4pt] \mathrm{P}(X=6) &= \frac{1}{6}+\left(\frac{1}{3}\right)\!\left(\frac{1}{6}\right) = \frac{2}{9} \end{aligned}\]

(b) \[\begin{aligned} \mathrm{E}(X) &= (1)\!\left(\frac{1}{18}\right)+(2)\!\left(\frac{1}{18}\right)+(3)\!\left(\frac{4}{18}\right)+(4)\!\left(\frac{4}{18}\right)+(5)\!\left(\frac{4}{18}\right)+(6)\!\left(\frac{4}{18}\right)\\[4pt] &= \frac{1+2+12+16+20+24}{18}\\[4pt] &= \frac{75}{18} = \frac{25}{6} \end{aligned}\]

(c) \[ \mathrm{E}(X^2) = \frac{349}{18}, \qquad \mathrm{E}(X) = \frac{25}{6} \]

Answer: (b) \(\mathrm{E}(X)=\tfrac{25}{6}\) (c) \(\mathrm{Var}(X)=\tfrac{73}{36}\)

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