| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Probability distribution from formula |
| Difficulty | Moderate -0.8 This is a straightforward S1 probability distribution question requiring only routine application of standard formulas: summing probabilities to find k, evaluating the cumulative distribution function, and calculating expectations. All steps are mechanical with no problem-solving or insight required, making it easier than average but not trivial due to the algebraic manipulation needed. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks |
|---|---|
| (a) \(\sum P(x) = k + \frac{1}{2}k + \frac{1}{4}k + \frac{1}{8}k = \frac{15}{8}k = 1 \therefore k = \frac{8}{15}\) | M2 A1 |
| (b) \(\frac{12}{25} + \frac{6}{25} = \frac{18}{25}\) | M1 A1 |
| (c) \(\sum xP(x) = \frac{12}{25} + \frac{12}{25} + \frac{12}{25} + \frac{12}{25} = \frac{48}{25}\) | M1 A1 |
| (d) \(E(X^2) = \sum x^2P(x) = \frac{12}{25} + \frac{24}{25} + \frac{36}{25} + \frac{48}{25} = \frac{24}{5}\); \(E(X^2+2) = \frac{24}{5} + 2 = \frac{34}{5}\) | M1 A1 M1 A1 |
| (11) |
**(a)** $\sum P(x) = k + \frac{1}{2}k + \frac{1}{4}k + \frac{1}{8}k = \frac{15}{8}k = 1 \therefore k = \frac{8}{15}$ | M2 A1 |
**(b)** $\frac{12}{25} + \frac{6}{25} = \frac{18}{25}$ | M1 A1 |
**(c)** $\sum xP(x) = \frac{12}{25} + \frac{12}{25} + \frac{12}{25} + \frac{12}{25} = \frac{48}{25}$ | M1 A1 |
**(d)** $E(X^2) = \sum x^2P(x) = \frac{12}{25} + \frac{24}{25} + \frac{36}{25} + \frac{48}{25} = \frac{24}{5}$; $E(X^2+2) = \frac{24}{5} + 2 = \frac{34}{5}$ | M1 A1 M1 A1 |
| | | (11) |
---2. The discrete random variable $X$ has the probability function shown below.
$$P ( X = x ) = \left\{ \begin{array} { l c }
\frac { k } { x } , & x = 1,2,3,4 \\
0 , & \text { otherwise } .
\end{array} \right.$$
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac { 12 } { 25 }$
Find
\item $\mathrm { F } ( 2 )$,
\item $\mathrm { E } ( X )$,
\item $\mathrm { E } \left( X ^ { 2 } + 2 \right)$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q2 [11]}}