| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | One unknown from sum constraint only |
| Difficulty | Moderate -0.8 This is a straightforward S1 question testing basic probability distribution properties. Part (a) uses the simple constraint that probabilities sum to 1 (one equation, one unknown). Parts (b)-(e) are direct applications of standard formulas with no problem-solving required. This is easier than average A-level maths questions as it's purely procedural recall from an introductory statistics module. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( X = x )\) | 0.1 | 0.35 | \(k\) | 0.15 | \(k\) |
| Answer | Marks |
|---|---|
| \(0.1 + 0.35 + k + 0.15 + k = 1\) | M1 |
| \(2k = 0.4; \quad k = 0.2\) | A1 |
| Answer | Marks |
|---|---|
| \(0.1 + 0.35 = 0.45\) | A1 |
| Answer | Marks |
|---|---|
| \(0.35 + 0.2 = 0.55\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(\Sigma xP(x) = 0.1 + 0.7 + 0.6 + 0.6 + 1 = 3\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X^2) = \Sigma x^2 P(x) = 0.1 + 1.4 + 1.8 + 2.4 + 5 = 10.7\) | M1 A1 | |
| \(\text{Var}(X) = 10.7 - 3^2 = 1.7\) | M1 | |
| \(\text{Var}(3X + 2) = 3^2 \times 1.7 = 15.3\) | M1 A1 | (12) |
**(a)**
$0.1 + 0.35 + k + 0.15 + k = 1$ | M1 |
$2k = 0.4; \quad k = 0.2$ | A1 |
**(b)**
$0.1 + 0.35 = 0.45$ | A1 |
**(c)**
$0.35 + 0.2 = 0.55$ | M1 A1 |
**(d)**
$\Sigma xP(x) = 0.1 + 0.7 + 0.6 + 0.6 + 1 = 3$ | M1 A1 |
**(e)**
$E(X^2) = \Sigma x^2 P(x) = 0.1 + 1.4 + 1.8 + 2.4 + 5 = 10.7$ | M1 A1 |
$\text{Var}(X) = 10.7 - 3^2 = 1.7$ | M1 |
$\text{Var}(3X + 2) = 3^2 \times 1.7 = 15.3$ | M1 A1 | (12)
---4. The discrete random variable $X$ has the following probability distribution.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$\mathrm { P } ( X = x )$ & 0.1 & 0.35 & $k$ & 0.15 & $k$ \\
\hline
\end{tabular}
\end{center}
Calculate
\begin{enumerate}[label=(\alph*)]
\item $k$,
\item $\mathrm { F } ( 2 )$,
\item $\mathrm { P } ( 1.3 < X \leq 3.8 )$,
\item $\mathrm { E } ( X )$,
\item $\operatorname { Var } ( 3 X + 2 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [12]}}