| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2018 |
| Session | June |
| Topic | Cumulative distribution functions |
| Type | PDF of transformed variable |
| Difficulty | Challenging +1.2 This is a standard Further Maths statistics question requiring differentiation of a CDF to find the PDF, then calculating E(X) by integration, and applying the transformation formula for Y=1/X². While it involves multiple steps and the transformation technique, these are well-practiced procedures in Further Maths with no novel insight required. The transformation is straightforward once the formula is recalled. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03g Cdf of transformed variables |
4 The continuous random variable $X$ has cumulative distribution function given by
$$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 8 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$$
(i) Find $\mathrm { E } ( X )$.\\
(ii) Find the probability density function of $Y$, where $Y = \frac { 1 } { X ^ { 2 } }$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2018 Q4}}