| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2002 |
| Session | November |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Multiple unknowns from expectation and variance |
| Difficulty | Moderate -0.8 This is a standard S1 textbook exercise testing routine application of discrete probability distribution formulas. Part (a) requires solving simultaneous equations using ΣP=1 and E(X) definition, parts (b-e) are direct formula applications with no problem-solving insight needed. The multi-part structure and 15 marks reflect thoroughness rather than difficulty. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
| \(P(X = x)\) | \(\alpha\) | \(0.2\) | \(0.1\) | \(0.2\) | \(\beta\) |
| Answer | Marks |
|---|---|
| \(\alpha + \beta = 0.5\) | B1 |
| \(-2\alpha + 2\beta = -0.2\) | M1 |
| \(\alpha = 0.3, \beta = 0.2\) | M1 A1; A1 |
| (6 marks) |
| Answer | Marks |
|---|---|
| \(F(0.8) = 0.6\) | B1 ft |
| (1 mark) |
| Answer | Marks |
|---|---|
| \(E(X^2) = (4 \times 0.3) + \ldots + (4 \times 0.2) = 2.4\) | M1, A1 |
| \(\text{Var}(X) = 2.4 - (-0.2)^2 = 2.36\) | M1, A1 |
| (4 marks) |
| Answer | Marks |
|---|---|
| \(E(3X - 2) = 3E(X) - 2 = -2.6\) | M1, A1 ft |
| (2 marks) |
| Answer | Marks |
|---|---|
| \(\text{Var}(2X + 6) = 4 \text{Var}(X) = 9.44\) | M1, A1 ft |
| (2 marks) |
## (a)
$\alpha + \beta = 0.5$ | B1 |
$-2\alpha + 2\beta = -0.2$ | M1 |
$\alpha = 0.3, \beta = 0.2$ | M1 A1; A1 |
| (6 marks) |
## (b)
$F(0.8) = 0.6$ | B1 ft |
| (1 mark) |
## (c)
$E(X^2) = (4 \times 0.3) + \ldots + (4 \times 0.2) = 2.4$ | M1, A1 |
$\text{Var}(X) = 2.4 - (-0.2)^2 = 2.36$ | M1, A1 |
| (4 marks) |
## (d)
$E(3X - 2) = 3E(X) - 2 = -2.6$ | M1, A1 ft |
| (2 marks) |
## (e)
$\text{Var}(2X + 6) = 4 \text{Var}(X) = 9.44$ | M1, A1 ft |
| (2 marks) |
**Total: 15 marks**
---The discrete random variable $X$ has the following probability distribution.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & $-2$ & $-1$ & $0$ & $1$ & $2$ \\
\hline
$P(X = x)$ & $\alpha$ & $0.2$ & $0.1$ & $0.2$ & $\beta$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Given that $E(X) = -0.2$, find the value of $\alpha$ and the value of $\beta$. [6]
\item Write down $F(0.8)$. [1]
\item Evaluate $\text{Var}(X)$. [4]
\end{enumerate}
Find the value of
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item $E(3X - 2)$, [2]
\item $\text{Var}(2X + 6)$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2002 Q6 [15]}}