CAIE FP2 2017 June — Question 9 12 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2017
SessionJune
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicExponential Distribution
TypeFind constant in PDF
DifficultyStandard +0.8 This question involves finding a normalizing constant (routine), recognizing an exponential distribution to state E(X) (standard), computing quartiles requiring logarithms (moderate), and most significantly, deriving a PDF through transformation Y=2^X using the Jacobian method (challenging for A-level). The transformation part requires careful application of the change-of-variable formula and is non-routine, elevating this above typical Further Maths statistics questions.
Spec5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03g Cdf of transformed variables
The continuous random variable \(X\) has probability density function f given by $$\text{f}(x) = \begin{cases} 0 & x < 0, \\ ae^{-x \ln 2} & x \geqslant 0, \end{cases}$$ where \(a\) is a positive constant.
  1. Find the value of \(a\). [2]
  2. State the value of E\((X)\). [1]
  3. Find the interquartile range of \(X\). [4]
The variable \(Y\) is related to \(X\) by \(Y = 2^X\).
  1. Find the probability density function of \(Y\). [5]
Question 9:
AnswerMarks Guidance
9(i)(a / ln 2) [– e–x ln 2] ∞ = a / ln 2 so a = ln 2 or 0⋅693
0M1 A1 State a or find a by equating ∫ ∞ f(x) dx to 1
0
AnswerMarks
Total:2
AnswerMarks Guidance
9(ii)E(X) = 1 / ln 2 or 1⋅44 B1
Total:1
QuestionAnswer Marks
AnswerMarks Guidance
9(iii)F(Q) = 1 – e–Q ln 2 = ¼ or ¾ M1
Q = (ln 4/3) / (ln 2) [= 0⋅415 ] (AEF)
AnswerMarks Guidance
1A1 Find one [lower] quartile Q
1
Q = (ln 4) / (ln 2) [= 2] (AEF)
AnswerMarks Guidance
3A1 Find other [upper] quartile Q
3
Q – Q [= (ln 3) / (ln 2)] = 1⋅58 [or 1⋅59]
AnswerMarks Guidance
3 1A1 Find interquartile range (allow Q – Q )
1 3
AnswerMarks
Total:4
AnswerMarks
9(iv)EITHER:
G(y) = P(Y < y) = P(2X < y)
= P(X < (ln y) / (ln 2))
= F((ln y) / (ln 2)) or F(log y) (AEF)
AnswerMarks Guidance
2(M1 A1 Find or state G(y) for x ⩾ 0 from Y = 2X
(allow < or ≤ throughout)
AnswerMarks
= 1 – e–ln y or 1 – 1/yA1)
OR:
AnswerMarks Guidance
Use x = (ln y) / (ln 2) to find both(M1 Find f(x) and dx/dy for use in g(y) = f(x) × dx/dy
f(x) = (ln 2) e–x ln 2 = (ln 2) e–ln y = (1/y) ln 2A1
and dx/dy = 1 / (y ln 2)A1)
g(y) [= G′(y)] = 1/y 2A1 Find g(y) in simplest form
for y ⩾ 1 [g(y) = 0 otherwise]A1 State corresponding range of y for G(y) or g(y)
Total:5 
Question 9:
--- 9(i) ---
9(i) | (a / ln 2) [– e–x ln 2] ∞ = a / ln 2 so a = ln 2 or 0⋅693
0 | M1 A1 | State a or find a by equating ∫ ∞ f(x) dx to 1
0
Total: | 2
--- 9(ii) ---
9(ii) | E(X) = 1 / ln 2 or 1⋅44 | B1 | State or find E(X)
Total: | 1
Question | Answer | Marks | Guidance
--- 9(iii) ---
9(iii) | F(Q) = 1 – e–Q ln 2 = ¼ or ¾ | M1 | Formulate equation for either quartile value Q
Q = (ln 4/3) / (ln 2) [= 0⋅415 ] (AEF)
1 | A1 | Find one [lower] quartile Q
1
Q = (ln 4) / (ln 2) [= 2] (AEF)
3 | A1 | Find other [upper] quartile Q
3
Q – Q [= (ln 3) / (ln 2)] = 1⋅58 [or 1⋅59]
3 1 | A1 | Find interquartile range (allow Q – Q )
1 3
Total: | 4
--- 9(iv) ---
9(iv) | EITHER:
G(y) = P(Y < y) = P(2X < y)
= P(X < (ln y) / (ln 2))
= F((ln y) / (ln 2)) or F(log y) (AEF)
2 | (M1 A1 | Find or state G(y) for x ⩾ 0 from Y = 2X
(allow < or ≤ throughout)
= 1 – e–ln y or 1 – 1/y | A1)
OR:
Use x = (ln y) / (ln 2) to find both | (M1 | Find f(x) and dx/dy for use in g(y) = f(x) × dx/dy
f(x) = (ln 2) e–x ln 2 = (ln 2) e–ln y = (1/y) ln 2 | A1
and dx/dy = 1 / (y ln 2) | A1)
g(y) [= G′(y)] = 1/y 2 | A1 | Find g(y) in simplest form
for y ⩾ 1 [g(y) = 0 otherwise] | A1 | State corresponding range of y for G(y) or g(y)
Total: | 5
The continuous random variable $X$ has probability density function f given by
$$\text{f}(x) = \begin{cases} 0 & x < 0, \\ ae^{-x \ln 2} & x \geqslant 0, \end{cases}$$
where $a$ is a positive constant.

\begin{enumerate}[label=(\roman*)]
\item Find the value of $a$. [2]
\item State the value of E$(X)$. [1]
\item Find the interquartile range of $X$. [4]
\end{enumerate}

The variable $Y$ is related to $X$ by $Y = 2^X$.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Find the probability density function of $Y$. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP2 2017 Q9 [12]}}
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