| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Multiple unknowns from expectation and variance |
| Difficulty | Standard +0.8 This question requires setting up and solving a system of three simultaneous equations (probabilities sum to 1, expectation formula, variance formula) with three unknowns. While the individual formulas are standard S1 content, coordinating all three equations and the algebraic manipulation to solve the system elevates this beyond routine exercises, requiring careful systematic work and algebraic fluency. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(x\) | - 3 | 0 | 2 | 4 |
| \(\mathrm { P } ( X = x )\) | \(p\) | \(q\) | \(r\) | 0.4 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \([a = 0.2 + 0.012t]\) | M1 | For differentiating to find \(a(t)\) |
| \([0.2 + 0.012t = 2.5 \times 0.2]\) | M1 | For attempting to solve \(a(t) = 2.5a(0)\) |
| \(t = 25\) | A1 | 3 marks total; AG |
| (ii) \([s = 0.1t^2 + 0.002t^3 \text{ (+ C)}]\) | M1 | For integrating to find \(s(t)\) |
| \([s = 0.1 \times 625 + 0.002 \times 15625]\) | DM1 | For using limits 0 to 25 or evaluating \(s(t)\) with \(C = 0\) (which may be implied by its absence) |
| Displacement is 93.75 (accept 93.7 or 93.8) | A1 | 3 marks total |
**(i)** $[a = 0.2 + 0.012t]$ | M1 | For differentiating to find $a(t)$
$[0.2 + 0.012t = 2.5 \times 0.2]$ | M1 | For attempting to solve $a(t) = 2.5a(0)$
$t = 25$ | A1 | 3 marks total; AG
**(ii)** $[s = 0.1t^2 + 0.002t^3 \text{ (+ C)}]$ | M1 | For integrating to find $s(t)$
$[s = 0.1 \times 625 + 0.002 \times 15625]$ | DM1 | For using limits 0 to 25 or evaluating $s(t)$ with $C = 0$ (which may be implied by its absence)
Displacement is 93.75 (accept 93.7 or 93.8) | A1 | 3 marks total
---2 The discrete random variable $X$ has the following probability distribution.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & - 3 & 0 & 2 & 4 \\
\hline
$\mathrm { P } ( X = x )$ & $p$ & $q$ & $r$ & 0.4 \\
\hline
\end{tabular}
\end{center}
Given that $\mathrm { E } ( X ) = 2.3$ and $\operatorname { Var } ( X ) = 3.01$, find the values of $p , q$ and $r$.
\hfill \mbox{\textit{CAIE S1 2012 Q2 [6]}}