| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2017 |
| Session | Specimen |
| Marks | 9 |
| Topic | Cumulative distribution functions |
| Type | CDF of transformed variable |
| Difficulty | Challenging +1.2 This is a Further Maths statistics question requiring transformation of random variables and improper integral analysis. Part (i) involves finding F_Y(y) through the relationship P(Y ≤ y) = P(1/X² ≤ y) = P(X ≥ 1/√y), which requires careful manipulation but follows a standard technique. Part (ii) requires recognizing that E(Y) = ∫(1/x²)f(x)dx diverges, involving differentiation of F to get f, then showing the integral is improper and divergent—conceptually more demanding than typical A-level but still a structured application of known methods. |
| Spec | 5.03d E(g(X)): general expectation formula5.03g Cdf of transformed variables |
The continuous random variable $X$ has cumulative distribution function given by
$$F(x) = \begin{cases}
0 & x < 0, \\
\frac{1}{16}x^2 & 0 \leq x \leq 4, \\
1 & x > 4.
\end{cases}$$
\begin{enumerate}[label=(\roman*)]
\item The random variable $Y$ is defined by $Y = \frac{1}{X^2}$. Find the cumulative distribution function of $Y$. [5]
\item Show that E$(Y)$ is not defined. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2017 Q9 [9]}}