| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2025 |
| Session | April |
| Marks | 9 |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find multiple parameters from system |
| Difficulty | Standard +0.3 This is a standard Further Maths Statistics question on continuous probability distributions requiring routine integration and algebraic manipulation. Part (a)(i) tests basic understanding of pdf support, (a)(ii) uses the integral condition ∫f(y)dy=1, part (b) applies E(Y) formula, and part (c) requires finding the maximum of a quadratic. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
The random variable $y$ has probability density function f(y) given by
$$f(y) = \begin{cases}
ky(a - y) & 0 \leq y \leq 3 \\
0 & \text{otherwise}
\end{cases}$$
where $k$ and $a$ are positive constants.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Explain why $a \geq 3$ [1]
\item Show that $k = \frac{2}{9(a - 2)}$ [3]
\end{enumerate}
\end{enumerate}
Given that $E(Y) = 1.75$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the values of a and k. [4]
\item Write down the mode of Y [1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2025 Q7 [9]}}