| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate E(X) from given distribution |
| Difficulty | Standard +0.3 This is a straightforward S3 question on expectation and bias requiring standard formulas. Part (a) is routine calculation of E(X), part (b) applies the definition of bias (expected value minus parameter), and part (c) uses the sample mean to estimate k. All steps are mechanical applications of learned techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02b Expectation and variance: discrete random variables |
| \(x\) | 2 | 4 | 7 | \(k\) |
| \(P(X = x)\) | 0.05 | 0.15 | 0.3 | 0.5 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(E(X) = (2 \times 0.05) + (4 \times 0.15) + (7 \times 0.3) + (k \times 0.5)\) | M1 | |
| \(= 2.8 + 0.5k\) | A1 | |
| (b) \(E(2\bar{X} − 5) = 2(2.8 + 0.5k) − 5 = k + 0.6\) | M1 | |
| \(\therefore\) bias = 0.6 | M1 A1 | |
| (c) unbiased est. of \(k = 2\bar{X} − 5.6 = (2 \times 8.34) − 5.6 = 11.08\) | M1 A1 | (7) |
**(a)** $E(X) = (2 \times 0.05) + (4 \times 0.15) + (7 \times 0.3) + (k \times 0.5)$ | M1 |
$= 2.8 + 0.5k$ | A1 |
**(b)** $E(2\bar{X} − 5) = 2(2.8 + 0.5k) − 5 = k + 0.6$ | M1 |
$\therefore$ bias = 0.6 | M1 A1 |
**(c)** unbiased est. of $k = 2\bar{X} − 5.6 = (2 \times 8.34) − 5.6 = 11.08$ | M1 A1 | (7)The discrete random variable $X$ has the probability distribution given below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$x$ & 2 & 4 & 7 & $k$ \\
\hline
$P(X = x)$ & 0.05 & 0.15 & 0.3 & 0.5 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the mean of $X$ in terms of $k$. [2 marks]
\item Find the bias in using $(2\overline{X} - 5)$ as an estimator of $k$. [3 marks]
\end{enumerate}
Fifty observations of $X$ were made giving a sample mean of 8.34 correct to 3 significant figures.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Calculate an unbiased estimate of $k$. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q3 [7]}}