CAIE S2 2024 November — Question 4 6 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2024
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicContinuous Probability Distributions and Random Variables
TypePiecewise PDF with k
DifficultyModerate -0.3 This is a standard two-part PDF question requiring routine integration techniques. Part (a) uses the fundamental property that the PDF integrates to 1, and part (b) applies the definition of expectation. Both involve straightforward polynomial integration with no conceptual challenges or novel problem-solving, making it slightly easier than average for A-level statistics.
Spec5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration
A random variable \(X\) has probability density function \(f\) defined by $$f(x) = \begin{cases} \frac{a}{x^2} - \frac{18}{x^3} & 2 \leqslant x < 3, \\ 0 & \text{otherwise}, \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = \frac{27}{2}\). [3]
  2. Show that \(\text{E}(X) = \frac{27}{2} \ln \frac{3}{2} - 3\). [3]
Question 4:
AnswerMarks
4(a)3
(a −18)dx = 1
x2 x3
AnswerMarks Guidance
2M1 Attempt integrate f(x), ignore limits and ‘= 1’.
3
−a + 9  = 1
AnswerMarks Guidance
 x x2 2A1 OE Correct integration and limits.
 −a +1+a −9  = 1 a = 2 7 (AG)
AnswerMarks Guidance
 3 2 4  2A1 Must see correct substitution of limits.
Correct working no errors seen.
3
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
4(b)3
(27 −18)dx
2x x2
AnswerMarks Guidance
2M1 Attempt to integrate xf(x), ignore limits.
3 3
  27lnx+18  or   27ln2x+18 
AnswerMarks Guidance
2 x 2 2 x 2A1 Correct integration and limits.
OE e.g. using ln 2x.
= 27ln3+ 6 − 27ln2 − 9 = 27ln3 − 3 AG
AnswerMarks Guidance
2 2 2 2A1 Must see correct substitution of limits.
Correct working no errors seen.
3
AnswerMarks Guidance
QuestionAnswer Marks 
Question 4:
--- 4(a) ---
4(a) | 3
(a −18)dx = 1
x2 x3
2 | M1 | Attempt integrate f(x), ignore limits and ‘= 1’.
3
−a + 9  = 1
 x x2 2 | A1 | OE Correct integration and limits.
 −a +1+a −9  = 1 a = 2 7 (AG)
 3 2 4  2 | A1 | Must see correct substitution of limits.
Correct working no errors seen.
3
Question | Answer | Marks | Guidance
--- 4(b) ---
4(b) | 3
(27 −18)dx
2x x2
2 | M1 | Attempt to integrate xf(x), ignore limits.
3 3
  27lnx+18  or   27ln2x+18 
2 x 2 2 x 2 | A1 | Correct integration and limits.
OE e.g. using ln 2x.
= 27ln3+ 6 − 27ln2 − 9 = 27ln3 − 3 AG
2 2 2 2 | A1 | Must see correct substitution of limits.
Correct working no errors seen.
3
Question | Answer | Marks | Guidance
A random variable $X$ has probability density function $f$ defined by

$$f(x) = \begin{cases}
\frac{a}{x^2} - \frac{18}{x^3} & 2 \leqslant x < 3, \\
0 & \text{otherwise},
\end{cases}$$

where $a$ is a constant.

\begin{enumerate}[label=(\alph*)]
\item Show that $a = \frac{27}{2}$. [3]

\item Show that $\text{E}(X) = \frac{27}{2} \ln \frac{3}{2} - 3$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE S2 2024 Q4 [6]}}
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