OCR Further Statistics 2018 March — Question 2 5 marks

Exam BoardOCR
ModuleFurther Statistics (Further Statistics)
Year2018
SessionMarch
Marks5
TopicCumulative distribution functions
TypeExpectation from CDF/PDF
DifficultyChallenging +1.2 This is a Further Maths statistics question requiring differentiation of the CDF to find the PDF, then computing E(√X) using integration by parts or substitution. While it involves multiple steps and careful algebraic manipulation, it's a standard textbook exercise testing routine techniques rather than requiring novel insight.
Spec5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration
2 A continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \frac { 1 } { x ^ { 3 } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find \(E ( \sqrt { } X )\).
AnswerMarks Guidance
\(f(x) = \frac{3}{x^4}\) \((x \geq 1)\)M1 Attempt to differentiate F(x)
A1
\(E(\sqrt{X}) = \int_1^{\infty} \sqrt{xf(x)}dx\)M1FT Limits wrong or omitted: M1A0
\(= \int_1^{\infty} 3x^{-3.5}dx\)A1FT FT on their 3 and power of \(x\)
\(= \frac{5}{3}\) or 1.2A1 Awrt 1.2 
$f(x) = \frac{3}{x^4}$ $(x \geq 1)$ | M1 | Attempt to differentiate F(x)
| A1 |

$E(\sqrt{X}) = \int_1^{\infty} \sqrt{xf(x)}dx$ | M1FT | Limits wrong or omitted: M1A0
$= \int_1^{\infty} 3x^{-3.5}dx$ | A1FT | FT on their 3 and power of $x$

$= \frac{5}{3}$ or 1.2 | A1 | Awrt 1.2 | BC
2 A continuous random variable $X$ has cumulative distribution function given by

$$\mathrm { F } ( x ) = \begin{cases} 1 - \frac { 1 } { x ^ { 3 } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

Find $E ( \sqrt { } X )$.

\hfill \mbox{\textit{OCR Further Statistics 2018 Q2 [5]}}
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