| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Session | Specimen |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find multiple parameters from system |
| Difficulty | Standard +0.3 This is a standard S2 question on continuous probability distributions requiring integration to find parameters. While it involves multiple steps (using ∫f(y)dy=1 and E(Y) formula), the techniques are routine: polynomial integration, algebraic manipulation, and interpreting a quadratic pdf. The 'explain why a≥3' part requires minimal insight (f(y)≥0 condition). Slightly easier than average due to straightforward calculus and algebra. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
\begin{enumerate}
\item The random variable $Y$ has probability density function $\mathrm { f } ( y )$ given by
\end{enumerate}
$$\mathrm { f } ( y ) = \left\{ \begin{array} { c c }
k y ( a - y ) & 0 \leqslant y \leqslant 3 \\
0 & \text { otherwise }
\end{array} \right.$$
where $k$ and $a$ are positive constants.\\
(a) (i) Explain why $a \geqslant 3$\\
(ii) Show that $k = \frac { 2 } { 9 ( a - 2 ) }$
Given that $\mathrm { E } ( Y ) = 1.75$\\
(b) show that $a = 4$ and write down the value of $k$.
For these values of $a$ and $k$,\\
(c) sketch the probability density function,\\
(d) write down the mode of $Y$.\\
\hfill \mbox{\textit{Edexcel S2 Q7 [15]}}