| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Maximum or minimum of uniforms |
| Difficulty | Standard +0.3 This is a structured S2 question on order statistics with extensive scaffolding. Parts (a)-(b) are trivial uniform probability calculations. The CDF is given, making (c) straightforward differentiation. Parts (d)-(f) are routine applications. Only part (g) requires recognizing P(Y>3) = 1-F(3), which is standard. The multi-part structure and given CDF make this easier than a typical A-level question. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03g Cdf of transformed variables |
6. The three independent random variables $A , B$ and $C$ each has a continuous uniform distribution over the interval $[ 0,5 ]$.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { P } ( A > 3 )$.
\item Find the probability that $A , B$ and $C$ are all greater than 3 .
The random variable $Y$ represents the maximum value of $A , B$ and $C$.
The cumulative distribution function of $Y$ is
$$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \\ \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{cases}$$
\item Find the probability density function of $Y$.
\item Sketch the probability density function of $Y$.
\item Write down the mode of $Y$.
\item Find $\mathrm { E } ( Y )$.
\item Find $\mathrm { P } ( Y > 3 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2009 Q6 [13]}}