| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Multiple observations or trials |
| Difficulty | Moderate -0.3 Parts (a)-(d) are direct formula applications for uniform distribution (E(X)=midpoint, Var(X)=range²/12, standard results). Part (e) requires binomial approximation to normal, which is S2-standard but involves multiple steps. Overall slightly easier than average due to routine nature of all components. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(E(X) = \frac{3-1}{2} = 1\) | B1cao | (1 mark) |
| (b) \(\text{Var}(X) = \frac{(3+1)^2}{12} = \frac{4}{3}\) or equivalent | M1A1 | (2 marks) |
| (c) \(E(X^2) = \frac{4}{3} + 1, = \frac{7}{3}\) or equivalent | M1, A1 | (2 marks) |
| (d) \(P(X < 1.4) = 0.6\) | B1cao | (1 mark) |
| (e) \(P(X < 0) = 0.25\); \(Y\) is number of values less than 0; \(Y \sim \text{Bin}(40, 0.25)\); \(P(Y \geq 10) = 1 - P(Y \leq 9) = 1 - 0.4395 = 0.5605\) | B1, M1A1, M1, A1 | (5 marks) |
**(a)** $E(X) = \frac{3-1}{2} = 1$ | B1cao | (1 mark)
**(b)** $\text{Var}(X) = \frac{(3+1)^2}{12} = \frac{4}{3}$ or equivalent | M1A1 | (2 marks)
**(c)** $E(X^2) = \frac{4}{3} + 1, = \frac{7}{3}$ or equivalent | M1, A1 | (2 marks)
**(d)** $P(X < 1.4) = 0.6$ | B1cao | (1 mark)
**(e)** $P(X < 0) = 0.25$; $Y$ is number of values less than 0; $Y \sim \text{Bin}(40, 0.25)$; $P(Y \geq 10) = 1 - P(Y \leq 9) = 1 - 0.4395 = 0.5605$ | B1, M1A1, M1, A1 | (5 marks)
**Notes:**
- (b) M1 $\frac{(3-1)^2}{12}$ or $\frac{(3+1)^2}{12}$ or $\frac{(3-(-1))^2}{12}$; A1 awrt 1.33.
- (c) M1 "their(b)" + ["their(a)"]² or $\int_{-1}^{4} \frac{x^2}{4} dx$; A1 awrt 2.33.
- (e) B1 For writing or using the probability of a negative = 0.25; M1 Writing or use of $B(40, p)$; A1 Writing or use of $B(40, 0.25)$; M1 Writing or using $1 - P(Y \leq 9)$; A1 awrt 0.561 or 0.560.
**[11 marks total]**
---The continuous random variable $X$ is uniformly distributed over the interval $[-1,3]$. Find
\begin{enumerate}[label=(\alph*)]
\item E($X$)
[1]
\item Var($X$)
[2]
\item E($X^2$)
[2]
\item P($X < 1.4$)
[1]
\end{enumerate}
A total of 40 observations of $X$ are made.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Find the probability that at least 10 of these observations are negative.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2011 Q3 [11]}}