| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.3 This is a standard S1 probability distribution question requiring routine application of formulas: using ΣP=1 and E(X) to form simultaneous equations, applying linearity of expectation, and calculating variance. While it involves multiple steps (7+3+3 marks), each step follows textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(x\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) |
| P\((X = x)\) | \(p\) | \(0.1\) | \(q\) | \(q\) | \(0.3\) | \(0.2\) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(4p + 13q + 4.7 = 6.7\) giving \(4p + 13q = 2\); \(p + 2q + 0.6 = 1\) giving \(p + 2q = 0.4\). Solve: \(p = 0.24, q = 0.08\) | M1 A1 M1 A1 M1 A1 A1 | |
| (b) \(E(2X + a) = 2E(X) + a = 13.4 + a\) giving \(a = -13.4\) | M1 A1 A1 | |
| (c) \(E(X^2) = 48.54\) and \(\text{Var}(X) = 48.54 - 6.7^2 = 3.65\) | M1 A1 A1 | Total: 13 marks |
(a) $4p + 13q + 4.7 = 6.7$ giving $4p + 13q = 2$; $p + 2q + 0.6 = 1$ giving $p + 2q = 0.4$. Solve: $p = 0.24, q = 0.08$ | M1 A1 M1 A1 M1 A1 A1 |
(b) $E(2X + a) = 2E(X) + a = 13.4 + a$ giving $a = -13.4$ | M1 A1 A1 |
(c) $E(X^2) = 48.54$ and $\text{Var}(X) = 48.54 - 6.7^2 = 3.65$ | M1 A1 A1 | Total: 13 marksThe discrete random variable $X$ takes only the values $4, 5, 6, 7, 8$ and $9$. The probabilities of these values are given in the table:
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$x$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ \\
\hline
P$(X = x)$ & $p$ & $0.1$ & $q$ & $q$ & $0.3$ & $0.2$ \\
\hline
\end{tabular}
It is known that E$(X) = 6.7$. Find
\begin{enumerate}[label=(\alph*)]
\item the values of $p$ and $q$, [7 marks]
\item the value of $a$ for which E$(2X + a) = 0$, [3 marks]
\item Var$(X)$. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q5 [13]}}