| Exam Board | SPS |
|---|---|
| Module | SPS FM Statistics (SPS FM Statistics) |
| Year | 2024 |
| Session | September |
| Marks | 13 |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Calculate and compare mean, median, mode |
| Difficulty | Standard +0.3 This is a standard Further Maths statistics question requiring routine application of pdf properties (integration to 1), expectation formula, and median calculation. All techniques are textbook exercises with no novel insight required, though it involves multiple steps across three parts. Slightly easier than average due to the straightforward algebraic manipulation. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
7.
The continuous random variable $X$ has the following probability density function:
$$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ and $b$ are constants.\\
(i) Show that $2 a + 2 b = 1$.\\
(ii) It is given that $\mathrm { E } ( X ) = \frac { 11 } { 9 }$. Use this information to find a second equation connecting $a$ and $b$, and hence find the values of $a$ and $b$.\\
(iii) Determine whether the median of $X$ is greater than, less than, or equal to $\mathrm { E } ( X )$.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM Statistics 2024 Q7 [13]}}