| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2001 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Waiting time applications |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard uniform distribution properties (parts a-c require only recall of formulas), basic binomial probability (part d), and a simple triangular distribution calculation (parts f-g). While multi-part, each component is routine with no novel problem-solving required—easier than average A-level maths. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p)5.02e Discrete uniform distribution5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(T < 0.2) = \underline{0.2}\) | B1 | (1) |
| (b) \(\mu = E(T) = \underline{0.5}\) | B1 | (1) |
| (c) \(E(T^2) = \int_0^1 \frac{1}{k} t^2\,dt = \left[\frac{t^3}{3}\right]_0^1\) | M1, A1 (dep) | |
| \(Var(T) = \frac{1}{3} - 0 - \mu^2 = \underline{\frac{1}{12}}\) | M1, A1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq 4) = \underline{0.6296}\) | M1 (identify Binomial), M1, A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Expect variance to be reduced | B1, B1 | (2) |
| (f) \(P(T < 0.2) = \int_0^{0.2} 4t\,dt = \left[4\frac{t^2}{2}\right]_0^{0.2} = 2(0.2)^2 - 0 = \underline{0.08}\) | M1, A1 c.s.o. | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(Y \sim B(75, 0.08) \approx Po(6)\), \(\lambda = 6\) | M1, A1 | |
| \(P(Y > 7) = 1 - P(Y \leq 7) = 1 - 0.7440 = \underline{0.256}\) | M1, A1 | (4) — (17) |
| S.C. Normal Approx: \(N(6, 5.52)\), \(\sqrt{6}\), \(\sqrt{5.52} \rightarrow (0.261 \sim 0.262)\) | M1, A1 | (if \(\frac{3}{4}\) only) |
## Question 7:
**(a)** $P(T < 0.2) = \underline{0.2}$ | B1 | **(1)**
**(b)** $\mu = E(T) = \underline{0.5}$ | B1 | **(1)**
**(c)** $E(T^2) = \int_0^1 \frac{1}{k} t^2\,dt = \left[\frac{t^3}{3}\right]_0^1$ | M1, A1 (dep) |
$Var(T) = \frac{1}{3} - 0 - \mu^2 = \underline{\frac{1}{12}}$ | M1, A1 | **(4)**
**(d)** $X$ = no. of children with $T < 0.2$, $X \sim B(20, 0.2)$
$P(X \leq 4) = \underline{0.6296}$ | M1 (identify Binomial), M1, A1 | **(3)**
**(e)** Expect mean to still be close to $0.5$ (or no change)
Expect variance to be **reduced** | B1, B1 | **(2)**
**(f)** $P(T < 0.2) = \int_0^{0.2} 4t\,dt = \left[4\frac{t^2}{2}\right]_0^{0.2} = 2(0.2)^2 - 0 = \underline{0.08}$ | M1, A1 c.s.o. | **(2)**
**(g)** $Y$ = no. of players stopping star in under 2s
$Y \sim B(75, 0.08) \approx Po(6)$, $\lambda = 6$ | M1, A1 |
$P(Y > 7) = 1 - P(Y \leq 7) = 1 - 0.7440 = \underline{0.256}$ | M1, A1 | **(4)** — **(17)**
**S.C.** Normal Approx: $N(6, 5.52)$, $\sqrt{6}$, $\sqrt{5.52} \rightarrow (0.261 \sim 0.262)$ | M1, A1 | (if $\frac{3}{4}$ only)7. In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key $T$ s after the star first appears, a simple model of the game assumes that $T$ is a continuous uniform random variable defined over the interval $[ 0,1 ]$.
\begin{enumerate}[label=(\alph*)]
\item Write down $\mathrm { P } ( \mathrm { T } < 0.2 )$.
\item Write down E(T).
\item Use integration to find $\operatorname { Var } ( T )$.
A group of 20 children each play this game once.
\item Find the probability that no more than 4 children stop the star in less than 0.2 s .
The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.
\item Explain how you would expect the mean and variance of T to change.
It is found that a more appropriate model of the game when played by experienced players assumes that $T$ has a probability density function $\mathrm { g } ( t )$ given by
$$g ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5 \\ 4 - 4 t , & 0.5 \leq t \leq 1 , \\ 0 , & \text { otherwise } . \end{cases}$$
\item Using this model show that $\mathrm { P } ( T < 0.2 ) = 0.08$.
A group of 75 experienced players each played this game once.
\item Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s .\\
(4)
END
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2001 Q7 [17]}}