| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Simple algebraic expression for P(X=x) |
| Difficulty | Moderate -0.3 This is a straightforward S1 probability distribution question requiring standard techniques: summing probabilities to find k, calculating P(X<0) by addition, and deriving a cumulative distribution function. While part (c) involves algebraic manipulation to show the quadratic form, all steps follow routine procedures with no novel problem-solving required. Slightly easier than average due to being a standard textbook-style exercise. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.03e Find cdf: by integration |
| Answer | Marks |
|---|---|
| (a) \(k(1 + 2 + \ldots + 8) = 1\) | M1 M1 A1 |
| \(36k = 1\), so \(k = \frac{1}{36}\) | |
| (b) \(P(X < 0) = \frac{5}{36} \times \frac{1}{6}\) | M1 A1 |
| (c) \(F(X) = \frac{1}{36} + \frac{2}{36} + \ldots + \frac{x+4}{36} = \frac{1}{36}(1 + 2 + \ldots [x+4])\) | M1 A1 |
| \(= \frac{1}{2} \times \frac{1}{36}(x+4)(x+5) + 1) = \frac{1}{72}(x+4)(x+5)\) | M1 M1 A1 A1 |
| 11 marks total |
(a) $k(1 + 2 + \ldots + 8) = 1$ | M1 M1 A1 |
$36k = 1$, so $k = \frac{1}{36}$ |
(b) $P(X < 0) = \frac{5}{36} \times \frac{1}{6}$ | M1 A1 |
(c) $F(X) = \frac{1}{36} + \frac{2}{36} + \ldots + \frac{x+4}{36} = \frac{1}{36}(1 + 2 + \ldots [x+4])$ | M1 A1 |
$= \frac{1}{2} \times \frac{1}{36}(x+4)(x+5) + 1) = \frac{1}{72}(x+4)(x+5)$ | M1 M1 A1 A1 |
| 11 marks total |The discrete random variable $X$ has probability function P$(X = x) = k(x + 4)$.
Given that $X$ can take any of the values $-3, -2, -1, 0, 1, 2, 3, 4$,
\begin{enumerate}[label=(\alph*)]
\item find the value of the constant $k$. [3 marks]
\item Find P$(X < 0)$. [2 marks]
\item Show that the cumulative distribution F$(x)$ is given by
$$\text{F}(x) = \lambda(x + 4)(x + 5)$$
where $\lambda$ is a constant to be found. [6 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [11]}}