| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Find cumulative distribution F(x) |
| Difficulty | Moderate -0.8 This is a straightforward S1 question requiring basic understanding of cumulative distribution functions. Part (a) uses the fact that F(4)=1 to find k (solving k×16=1), and part (b) involves simple subtraction to convert from CDF to PMF. It's routine recall and mechanical calculation with no problem-solving insight required, making it easier than average. |
| Spec | 2.04a Discrete probability distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(F(4) = 1\) | M1 | For writing or using \(F(4)=1\) |
| \(4^2k = 1 \Rightarrow k = \frac{1}{16}\) | A1 | \(\frac{1}{16}\) or 0.0625; answer only scores 2 out of 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| e.g. \(P(X=2) = F(2) - F(1) = \frac{4}{16} - \frac{1}{16} = \frac{3}{16}\) | M1 | Correct use of \(F(x)\) to find \(P(X=2,3,\) or \(4)\); may be implied by at least two correct probabilities |
| Full table: \([x]\): 1, 2, 3, 4; \(P(X=x)\): \(\frac{1}{16}, \frac{3}{16}, \frac{5}{16}, \frac{7}{16}\) (0.0625, 0.1875, 0.3125, 0.4375) | A1 | Fully correct probability distribution; allow exact decimals; ignore incorrect/missing labels if table is all correct |
## Question 1:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $F(4) = 1$ | M1 | For writing or using $F(4)=1$ |
| $4^2k = 1 \Rightarrow k = \frac{1}{16}$ | A1 | $\frac{1}{16}$ or 0.0625; answer only scores 2 out of 2 |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| e.g. $P(X=2) = F(2) - F(1) = \frac{4}{16} - \frac{1}{16} = \frac{3}{16}$ | M1 | Correct use of $F(x)$ to find $P(X=2,3,$ or $4)$; may be implied by at least two correct probabilities |
| Full table: $[x]$: 1, 2, 3, 4; $P(X=x)$: $\frac{1}{16}, \frac{3}{16}, \frac{5}{16}, \frac{7}{16}$ (0.0625, 0.1875, 0.3125, 0.4375) | A1 | Fully correct probability distribution; allow exact decimals; ignore incorrect/missing labels if table is all correct |
---\begin{enumerate}
\item The discrete random variable $X$ can only take the values $1,2,3$ and 4 For these values the cumulative distribution function is defined by
\end{enumerate}
$$\mathrm { F } ( x ) = k x ^ { 2 } \text { for } x = 1,2,3,4$$
where $k$ is a constant.\\
(a) Find the value of $k$.\\
(b) Find the probability distribution of $X$.\\
\hfill \mbox{\textit{Edexcel S1 2015 Q1 [4]}}