| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Moderate -0.8 This is a straightforward S1 probability distribution question requiring standard techniques: summing probabilities to find k, calculating E(X) using the definition, applying probability and expectation rules, and computing variance. All parts follow textbook procedures with no problem-solving insight needed, making it easier than average but not trivial due to the arithmetic involved across multiple parts. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | M1 A1 | \(2k + 3k + 4k + 5k + 6k = 1\); \(k = \frac{1}{30}\) |
| (b) | M2 A1 | \(\sum xP(x) = \frac{1}{30}(4 + 9 + 16 + 25 + 36) = \frac{9}{2}\) |
| (c) | M1 A1 | \(= \mathrm{P}(X > \frac{9}{2}) = \frac{5}{20} + \frac{6}{20} = \frac{11}{20}\) |
| (d) | M1 A1 | \((2 \times \frac{9}{2}) - 5 = 4\) |
| (e) | M1 A1 | \(E(X^2) = \sum x^2P(x) = \frac{1}{30}(8 + 27 + 64 + 125 + 216) = 22\) |
| M1 A1, (13) | \(\mathrm{Var}(X) = 22 - (\frac{9}{2})^2 = \frac{7}{4}\) |
(a) | M1 A1 | $2k + 3k + 4k + 5k + 6k = 1$; $k = \frac{1}{30}$
(b) | M2 A1 | $\sum xP(x) = \frac{1}{30}(4 + 9 + 16 + 25 + 36) = \frac{9}{2}$
(c) | M1 A1 | $= \mathrm{P}(X > \frac{9}{2}) = \frac{5}{20} + \frac{6}{20} = \frac{11}{20}$
(d) | M1 A1 | $(2 \times \frac{9}{2}) - 5 = 4$
(e) | M1 A1 | $E(X^2) = \sum x^2P(x) = \frac{1}{30}(8 + 27 + 64 + 125 + 216) = 22$
| M1 A1, (13) | $\mathrm{Var}(X) = 22 - (\frac{9}{2})^2 = \frac{7}{4}$
---The discrete random variable $X$ has the probability function shown below.
$$P(X = x) = \begin{cases} kx, & x = 2, 3, 4, 5, 6, \\ 0, & \text{otherwise}. \end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $k$. [2 marks]
\item Show that E$(X) = \frac{9}{2}$. [3 marks]
\end{enumerate}
Find
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item P$[X > \text{E}(X)]$, [2 marks]
\item E$(2X - 5)$, [2 marks]
\item Var$(X)$. [4 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q5 [13]}}