| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Discrete CDF table with constants |
| Difficulty | Moderate -0.3 This is a straightforward S1 question testing basic CDF properties. Part (a) uses F(4)=1 to find k (simple algebra), part (b) converts CDF to PDF by subtraction, part (c) identifies the mode by inspection, and part (d) applies the standard variance transformation rule Var(aX+b)=a²Var(X). All steps are routine textbook procedures with no problem-solving insight required, making it slightly easier than average. |
| Spec | 2.04a Discrete probability distributions5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 1 | 2 | 3 | 4 |
| \(\mathrm {~F} ( x )\) | \(\frac { 1 } { 13 }\) | \(\frac { 2 k - 1 } { 26 }\) | \(\frac { 3 ( k + 1 ) } { 26 }\) | \(\frac { k + 4 } { 8 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{k+4}{8} = 1\) leading to \([k = 4]\) | B1* | Allow verification method \(\frac{4+4}{8} = 1\) provided they conclude \(k = 4\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=1) = \frac{1}{13}\), \(P(X=2) = \frac{7}{26} - \frac{1}{13} = \frac{5}{26}\), \(P(X=3) = \frac{15}{26} - \frac{7}{26} = \frac{4}{13}\), \(P(X=4) = 1 - \frac{15}{26} = \frac{11}{26}\) | M1 M1 A1 | First M1: one correct probability from \(x=2,3,4\); Second M1: second correct probability; A1: fully correct distribution |
| Answer | Marks | Guidance |
|---|---|---|
| \(4\) | B1ft | Must be consistent with highest probability in their distribution from (b). If no distribution found, answer must be 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = 1\times\frac{1}{13} + 2\times\frac{5}{26} + 3\times\frac{4}{13} + 4\times\frac{11}{26} = \frac{40}{13}\) | M1 | Correct method for \(E(X)\); use of \(\sum xF(x)\) is M0 |
| \(E(X^2) = 1^2\times\frac{1}{13} + 2^2\times\frac{5}{26} + 3^2\times\frac{4}{13} + 4^2\times\frac{11}{26} = \frac{135}{13}\) | M1 | Correct method for \(E(X^2)\); use of \(\sum x^2 F(x)\) is M0 |
| \(\text{Var}(X) = \frac{135}{13} - \left(\frac{40}{13}\right)^2 = \frac{155}{169}\) | M1 | Use of \(E(X^2) - E(X)^2\) |
| \(\text{Var}(13X-6) = 13^2 \times \frac{155}{169} = 155\) | M1 | Use of \(13^2\text{Var}(X)\) |
| \(= 155\) | A1 | cao |
## Question 7:
### Part (a):
| $\frac{k+4}{8} = 1$ leading to $[k = 4]$ | B1* | Allow verification method $\frac{4+4}{8} = 1$ provided they conclude $k = 4$ |
### Part (b):
| $P(X=1) = \frac{1}{13}$, $P(X=2) = \frac{7}{26} - \frac{1}{13} = \frac{5}{26}$, $P(X=3) = \frac{15}{26} - \frac{7}{26} = \frac{4}{13}$, $P(X=4) = 1 - \frac{15}{26} = \frac{11}{26}$ | M1 M1 A1 | First M1: one correct probability from $x=2,3,4$; Second M1: second correct probability; A1: fully correct distribution |
### Part (c):
| $4$ | B1ft | Must be consistent with highest probability in their distribution from (b). If no distribution found, answer must be 4 |
### Part (d):
| $E(X) = 1\times\frac{1}{13} + 2\times\frac{5}{26} + 3\times\frac{4}{13} + 4\times\frac{11}{26} = \frac{40}{13}$ | M1 | Correct method for $E(X)$; use of $\sum xF(x)$ is M0 |
| $E(X^2) = 1^2\times\frac{1}{13} + 2^2\times\frac{5}{26} + 3^2\times\frac{4}{13} + 4^2\times\frac{11}{26} = \frac{135}{13}$ | M1 | Correct method for $E(X^2)$; use of $\sum x^2 F(x)$ is M0 |
| $\text{Var}(X) = \frac{135}{13} - \left(\frac{40}{13}\right)^2 = \frac{155}{169}$ | M1 | Use of $E(X^2) - E(X)^2$ |
| $\text{Var}(13X-6) = 13^2 \times \frac{155}{169} = 155$ | M1 | Use of $13^2\text{Var}(X)$ |
| $= 155$ | A1 | cao |
---\begin{enumerate}
\item The cumulative distribution of a discrete random variable $X$ is given by
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 \\
\hline
$\mathrm {~F} ( x )$ & $\frac { 1 } { 13 }$ & $\frac { 2 k - 1 } { 26 }$ & $\frac { 3 ( k + 1 ) } { 26 }$ & $\frac { k + 4 } { 8 }$ \\
\hline
\end{tabular}
\end{center}
where $k$ is a positive constant.\\
(a) Show that $k = 4$\\
(b) Find the probability distribution of the discrete random variable $X$\\
(c) Using your answer to part (b), write down the mode of $X$\\
(d) Calculate $\operatorname { Var } ( 13 X - 6 )$
\hfill \mbox{\textit{Edexcel S1 2024 Q7 [10]}}