| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | December |
| Marks | 11 |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find parameter from expectation |
| Difficulty | Standard +0.8 Part (a) is routine Further Maths integration to find E(X) and solve for a parameter. Part (b) requires knowledge of order statistics (finding the CDF of the maximum of independent observations), which is a more advanced Further Statistics topic requiring F_M(x) = [F_X(x)]^n - this is non-trivial for A-level and goes beyond standard techniques. |
| 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 |
| Answer | Marks |
|---|---|
| M1, B1, A1 | (no additional guidance) |
| Answer | Marks |
|---|---|
| M1, A1ft, M1, M1, A1, A1, A1, A1 | Find F(x): \(= \frac{x^2}{a^2}\); FT on their a; Correct basis for CDF of m; Allow </≤ throughout; Correct function, any letter; Range 0 ≤ m ≤ 6; Letter not x, and 0, 1 present |
## (a)
$\int_0^a x \frac{2x}{a^2} dx = 4$
$\left[\frac{2x^3}{3a^2}\right] = 4$
$\frac{2}{3}a = 4 \Rightarrow a = 6$
| M1, B1, A1 | (no additional guidance) |
## (b)
$F(x) = \frac{x^2}{36}$
Let the CDF of $M$ be H(m). Then
$H(m) = P(\text{all observations less than } m)$
$= [P(X < m)]^5$
$= \left[\frac{m^2}{36}\right]^5$
$$H(m) = \begin{cases} 0 & m < 0, \\ \frac{m^{10}}{60466176} & 0 \leq m \leq 6, \\ 1 & m > 6. \end{cases}$$
| M1, A1ft, M1, M1, A1, A1, A1, A1 | Find F(x): $= \frac{x^2}{a^2}$; FT on their a; Correct basis for CDF of m; Allow </≤ throughout; Correct function, any letter; Range 0 ≤ m ≤ 6; Letter not x, and 0, 1 present |8 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 $M$.
\section*{OCR}
Oxford Cambridge and RSA
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q8 [11]}}