| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Uniform Random Variables |
| Type | Calculate probabilities and expectations |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard formulas for uniform distributions. Parts (a)-(c) require direct application of E(X) = (a+b)/2 and linearity of expectation, part (b) uses the uniform pdf, and part (d) applies the variance formula. All steps are routine with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(E(3 - 2X) = 3 - 2E(X) = [3 - 2(\frac{(a+b)}{2})]\) \(= 3 - a - b\) | M1, A1 | M1 for using \(3 - 2E(X)\) where \(E(X)\) is a linear function of \(a\) and \(b\). A1 for \(3 - a - b\) or \(3 - (a+b)\) |
| (b) \(P(X > \frac{1}{2}b + \frac{2}{3}a) = \frac{b - (\frac{1}{2}b + \frac{2}{3}a)}{b - a} = \frac{2}{3}\) OR \(1 - \frac{(\frac{1}{2}b + \frac{2}{3}a) - a}{b - a} = \frac{2}{3}\) | M1, A1 | M1 for a correct fraction expression for \(P(X > \frac{1}{2}b + \frac{2}{3}a)\) in terms of \(a\) and \(b\) (need brackets!). A1 for \(\frac{2}{3}\) |
| (c) \(E([3]X^2) = \int_a^b \frac{1}{b-a} [3]x^2 dx\) | M1 | 1st M1 for correct integral for \(E(3X^2)\) or \(E(X^2)\) (ignore limits) |
| \(= \left[\frac{1}{(b-a)}x^3\right]_a^b = \left(\frac{b^3 - (b')}{2b}\right)\) \(= b^2\) | dM1 | 2nd dM1 dependent on 1st M1 for correct integration and correct use of \(a = -b\) including in limits. Must be \(E(3X^2)\) |
| OR \([Var(X) = E(X^2) - [E(X)]^2]\) with \(\frac{(b-a)^2}{12} = E(X^2) - 0^2\) or \(E(X^2) - \frac{(a+b)^2}{4}\) | ||
| \(E(3X^2) = 3\left(\frac{(b-b)''}{12}\right)\) \(= b^2\) | A1 | A1 |
| (d) Range = \(b - a = 18\) or \(b - -b = 18\) or \(b = 9\) | M1, A1 | M1 for writing or using \((b - a) = 18\) or \(b - -b = 18\) or \(b = 9\). A1 for 27. [Correct answer only is 2/2] |
| \(\text{Var}(X) = \left[\frac{18^2}{12} \text{ or } \frac{9^2}{3} - 0^2\right] = 27\) |
**(a)** $E(3 - 2X) = 3 - 2E(X) = [3 - 2(\frac{(a+b)}{2})]$ $= 3 - a - b$ | M1, A1 | M1 for using $3 - 2E(X)$ where $E(X)$ is a linear function of $a$ and $b$. A1 for $3 - a - b$ or $3 - (a+b)$
**(b)** $P(X > \frac{1}{2}b + \frac{2}{3}a) = \frac{b - (\frac{1}{2}b + \frac{2}{3}a)}{b - a} = \frac{2}{3}$ OR $1 - \frac{(\frac{1}{2}b + \frac{2}{3}a) - a}{b - a} = \frac{2}{3}$ | M1, A1 | M1 for a correct fraction expression for $P(X > \frac{1}{2}b + \frac{2}{3}a)$ in terms of $a$ and $b$ (need brackets!). A1 for $\frac{2}{3}$
**(c)** $E([3]X^2) = \int_a^b \frac{1}{b-a} [3]x^2 dx$ | M1 | 1st M1 for correct integral for $E(3X^2)$ or $E(X^2)$ (ignore limits)
| $= \left[\frac{1}{(b-a)}x^3\right]_a^b = \left(\frac{b^3 - (b')}{2b}\right)$ $= b^2$ | dM1 | 2nd dM1 dependent on 1st M1 for correct integration and correct use of $a = -b$ including in limits. Must be $E(3X^2)$
| OR $[Var(X) = E(X^2) - [E(X)]^2]$ with $\frac{(b-a)^2}{12} = E(X^2) - 0^2$ or $E(X^2) - \frac{(a+b)^2}{4}$ | |
| $E(3X^2) = 3\left(\frac{(b-b)''}{12}\right)$ $= b^2$ | A1 | A1
**(d)** Range = $b - a = 18$ or $b - -b = 18$ or $b = 9$ | M1, A1 | M1 for writing or using $(b - a) = 18$ or $b - -b = 18$ or $b = 9$. A1 for 27. [Correct answer only is 2/2]
| $\text{Var}(X) = \left[\frac{18^2}{12} \text{ or } \frac{9^2}{3} - 0^2\right] = 27$ | |
**Total: 9**7. The continuous random variable $X$ is uniformly distributed over the interval $[ a , b ]$
\begin{enumerate}[label=(\alph*)]
\item Find an expression, in terms of $a$ and $b$, for $\mathrm { E } ( 3 - 2 X )$
\item Find $\mathrm { P } \left( X > \frac { 1 } { 3 } b + \frac { 2 } { 3 } a \right)$
Given that $\mathrm { E } ( X ) = 0$
\item find an expression, in terms of $b$ only, for $\mathrm { E } \left( 3 X ^ { 2 } \right)$
Given also that the range of $X$ is 18
\item find $\operatorname { Var } ( X )$\\
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{1a1534ea-4c62-4945-850a-9460ea87fd64-24_2630_1828_121_121}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2017 Q7 [9]}}