| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2021 |
| Session | June |
| Marks | 12 |
| Topic | Continuous Probability Distributions and Random Variables |
| Difficulty | Standard +0.8 This is a Further Maths Statistics question requiring multiple PDF properties: using both the normalization condition (∫f=1) and a given probability (P(X≥2)=0.75) to find two unknowns, then computing expectation and conditional probability. While the integration is straightforward, setting up and solving the simultaneous equations, plus correctly handling the conditional probability in part (iii), requires solid understanding beyond routine application. This is moderately challenging for FM level. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03d E(g(X)): general expectation formula |
A continuous random variable $X$ has probability density function $f$ given by
$$f(x) = \begin{cases}
\frac{x^2}{a} + b, & 0 \leq x \leq 4 \\
0 & \text{otherwise}
\end{cases}$$
where $a$ and $b$ are positive constants. It is given that $P(X \geq 2) = 0.75$.
\begin{enumerate}[label=(\roman*)]
\item Show that $a = 32$ and $b = \frac{1}{12}$. [5]
\item Find $E(X)$. [3]
\item Find $P(X > E(X)|X > 2)$ [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Statistics 2021 Q7 [12]}}