| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 10 |
| 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 systematic application of two fundamental properties: probabilities sum to 1 and the expectation formula. While it involves solving simultaneous equations with two unknowns, the algebraic manipulation is straightforward and the approach is routine for this topic. Slightly easier than average due to the mechanical nature of the solution. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( X = x )\) | \(a\) | \(b\) | \(\frac { 1 } { 4 }\) | \(2 a\) | \(\frac { 1 } { 8 }\) |
| Answer | Marks |
|---|---|
| \(a + b + \frac{1}{4} + 2a + \frac{1}{8} = 1\) | M1 |
| \(3a + b = \frac{5}{8}\); \(b = \frac{5}{8} - 3a\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(\sum xP(x) = a + 2b + \frac{1}{4} + 8a + \frac{5}{8}\) | M1 |
| \(= 9a + 2(\frac{5}{8} - 3a) + \frac{11}{8} = 3a + \frac{21}{8}\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(3a + \frac{21}{8} = \frac{45}{16}\) | M1 |
| \(3a = \frac{45}{16} - \frac{21}{8} = \frac{3}{16}\) | M1 |
| \(a = \frac{1}{16}, b = \frac{7}{16}\) | A2 |
| (10) |
**Part (a)**
$a + b + \frac{1}{4} + 2a + \frac{1}{8} = 1$ | M1 |
$3a + b = \frac{5}{8}$; $b = \frac{5}{8} - 3a$ | M1 A1 |
**Part (b)**
$\sum xP(x) = a + 2b + \frac{1}{4} + 8a + \frac{5}{8}$ | M1 |
$= 9a + 2(\frac{5}{8} - 3a) + \frac{11}{8} = 3a + \frac{21}{8}$ | M1 A1 |
**Part (c)**
$3a + \frac{21}{8} = \frac{45}{16}$ | M1 |
$3a = \frac{45}{16} - \frac{21}{8} = \frac{3}{16}$ | M1 |
$a = \frac{1}{16}, b = \frac{7}{16}$ | A2 |
| (10) |2. 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
$\mathrm { P } ( X = x )$ & $a$ & $b$ & $\frac { 1 } { 4 }$ & $2 a$ & $\frac { 1 } { 8 }$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $b$ in terms of $a$.
\item Find an expression for $\mathrm { E } ( X )$ in terms of $a$.
Given that $\mathrm { E } ( X ) = \frac { 45 } { 16 }$,
\item find the values of $a$ and $b$,
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q2 [10]}}