| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2026 |
| Session | January |
| Marks | 9 |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find parameter from expectation |
| Difficulty | Standard +0.3 This question involves standard techniques: (a) finding a parameter using E(X) = ∫xf(x)dx with a triangular distribution requires straightforward integration and algebra; (b) finding the CDF of the maximum of independent observations uses the standard formula F_M(m) = [F_X(m)]^n. Both parts are textbook exercises with no novel insight required, making this slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03g Cdf of transformed variables |
9 A continuous random variable $X$ has probability density function given by the following function, - where $a$ is a constant.\\
$\mathrm { f } ( x ) = \left\{ \begin{array} { l l } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a , \\ 0 & \text { otherwise. } \end{array} \right\}$\\
The expected value of $X$ is 4 .
\begin{enumerate}[label=(\alph*)]
\item Show that $a = 6$.
Five independent observations of $X$ are obtained, and the largest of them is denoted by $M$.
\item Find the cumulative distribution function of $M$.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2026 Q9 [9]}}