2021 H2 Mathematics Paper 2 Question 6

Discrete Random Variables (DRVs)

Answers

2.7{2.7}

Full solutions

∑P(X=x)=10.2+0.3+p+p+q=1\begin{align*} \sum P(X=x) &= 1 \\ 0.2+0.3+p+p+q &= 1 \end{align*}
q=0.5−2p\begin{equation} q = 0.5 - 2p \end{equation}
E(X)=∑xxP(X=x)=0.2+0.6+3p+4p+5q=0.8+7p+5q\begin{align*} \mathrm{E}(X) &= \sum_x x \mathrm{P}(X=x) \\ &= 0.2+0.6+3p+4p+5q \\ &= 0.8 + 7p + 5q \end{align*}
E(X2)=∑xx2P(X=x)=0.2+4(0.3)+9p+16p+25q=1.4+25p+25q\begin{align*} \mathrm{E}(X^2) &= \sum_x x^2 \mathrm{P}(X=x) \\ &= 0.2+4(0.3)+9p+16p+25q \\ &= 1.4 + 25p + 25q \\ \end{align*}
Var(X)=1.61E(X2)−(E(X))2=1.611.4+25p+25q−(0.8+7p+5q)2=1.61\begin{gather*} \mathrm{Var}(X) = 1.61 \\ \mathrm{E}(X^2) - \left(\mathrm{E}(X)\right)^2 = 1.61 \\ 1.4 + 25p + 25q - (0.8 + 7p + 5q)^2 = 1.61 \\ \end{gather*}
Substituting (1),{(1),}
1.4+25p+25(0.5−2p)−(0.8+7p+5(0.5−2p))2=1.611.4 + 25p + 25(0.5-2p) - (0.8+7p+5(0.5-2p))^2 = 1.61
Solving with a GC,
p=0.2p=0.2
q=0.5−2p=0.1\begin{align*} q &= 0.5 - 2p \\ &= 0.1 \end{align*}
Mean score=0.8+7p+5q=2.7  ■\begin{align*} &\textrm{Mean score}\\ &= 0.8+7p+5q\\ &= 2.7\;\blacksquare \end{align*}