Edexcel S2 2013 June — Question 4 9 marks

Exam BoardEdexcel
ModuleS2 (Statistics 2)
Year2013
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicContinuous Uniform Random Variables
TypeDerive or verify variance formula
DifficultyModerate -0.3 This is a straightforward S2 question testing standard uniform distribution properties. Part (a) requires recalling E(X) = (a+b)/2, part (b) is a guided integration using Var(X) = E(X²) - [E(X)]² with explicit instruction to 'show that', parts (c-e) apply standard formulas for variance of linear transformations, CDF definition, and median. All steps are routine applications of bookwork with no problem-solving or novel insight required, making it slightly easier than average.
Spec5.02e Discrete uniform distribution5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles5.04a Linear combinations: E(aX+bY), Var(aX+bY)
  1. A continuous random variable \(X\) is uniformly distributed over the interval [ \(b , 4 b\) ] where \(b\) is a constant.
    1. Write down \(\mathrm { E } ( X )\).
    2. Use integration to show that \(\operatorname { Var } ( X ) = \frac { 3 b ^ { 2 } } { 4 }\).
    3. Find \(\operatorname { Var } ( 3 - 2 X )\).
    Given that \(b = 1\) find
  2. the cumulative distribution function of \(X , \mathrm {~F} ( x )\), for all values of \(x\),
  3. the median of \(X\).
Question 4:
Part 4(a):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(E(X) = \frac{5b}{2}\)B1
Part 4(b):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\text{Var}(X) = E(X^2) - (E(X))^2\)M1 Using \(\int \frac{x^2}{3b}\,dx - (\text{their }(a))^2\); limits not needed
\(= \int_b^{4b} \frac{x^2}{3b}\,dx - \left(\frac{5b}{2}\right)^2\)M1d Dependent on previous M; correct integration \(x^n \to x^{n+1}\) and correct limits
\(= \left[\frac{x^3}{9b}\right]_b^{4b} - \frac{25b^2}{4}\)
\(= \frac{63b^3}{9b} - \frac{25b^2}{4}\)
\(= \frac{3b^2}{4}\)A1cso Correct solution with no incorrect working; answer given so must show working
Part 4(c):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\text{Var}(3 - 2X) = 4\text{Var}(X)\)M1 Writing or using \(4\text{Var}(X)\)
\(= 3b^2\)A1
Part 4(d):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(F(x) = \begin{cases} 0 & x < 1 \\ \frac{x-1}{3} & 1 \leq x \leq 4 \\ 1 & x > 4 \end{cases}\)B1B1 1st B1: top and bottom line (allow \(\leq\) instead of \(<\) and \(\geq\) instead of \(>\)); 2nd B1: middle row (allow \(<\) instead of \(\leq\))
Part 4(e):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\frac{x-1}{3} = 0.5\) so \(x = 2.5\)B1 
# Question 4:

## Part 4(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(X) = \frac{5b}{2}$ | B1 | |

## Part 4(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{Var}(X) = E(X^2) - (E(X))^2$ | M1 | Using $\int \frac{x^2}{3b}\,dx - (\text{their }(a))^2$; limits not needed |
| $= \int_b^{4b} \frac{x^2}{3b}\,dx - \left(\frac{5b}{2}\right)^2$ | M1d | Dependent on previous M; correct integration $x^n \to x^{n+1}$ and correct limits |
| $= \left[\frac{x^3}{9b}\right]_b^{4b} - \frac{25b^2}{4}$ | | |
| $= \frac{63b^3}{9b} - \frac{25b^2}{4}$ | | |
| $= \frac{3b^2}{4}$ | A1cso | Correct solution with no incorrect working; answer given so must show working |

## Part 4(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{Var}(3 - 2X) = 4\text{Var}(X)$ | M1 | Writing or using $4\text{Var}(X)$ |
| $= 3b^2$ | A1 | |

## Part 4(d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $F(x) = \begin{cases} 0 & x < 1 \\ \frac{x-1}{3} & 1 \leq x \leq 4 \\ 1 & x > 4 \end{cases}$ | B1B1 | 1st B1: top and bottom line (allow $\leq$ instead of $<$ and $\geq$ instead of $>$); 2nd B1: middle row (allow $<$ instead of $\leq$) |

## Part 4(e):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{x-1}{3} = 0.5$ so $x = 2.5$ | B1 | |

---
\begin{enumerate}
  \item A continuous random variable $X$ is uniformly distributed over the interval [ $b , 4 b$ ] where $b$ is a constant.\\
(a) Write down $\mathrm { E } ( X )$.\\
(b) Use integration to show that $\operatorname { Var } ( X ) = \frac { 3 b ^ { 2 } } { 4 }$.\\
(c) Find $\operatorname { Var } ( 3 - 2 X )$.
\end{enumerate}

Given that $b = 1$ find\\
(d) the cumulative distribution function of $X , \mathrm {~F} ( x )$, for all values of $x$,\\
(e) the median of $X$.\\

\hfill \mbox{\textit{Edexcel S2 2013 Q4 [9]}}
This paper (7 questions)
View full paper
Q1 11 Q2 10 Q3 13 Q4 9 Q5 12 Q6 10 Q7 10