| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find mode of distribution |
| Difficulty | Standard +0.3 This is a standard S2 question testing routine techniques: sketching a piecewise pdf, finding the mode by differentiation, calculating E(Y²) by integration, using Var(Y) = E(Y²) - [E(Y)]², and solving P(Y ≥ y) = 0.9 by integration. While multi-part with several steps, each component is a textbook exercise requiring only direct application of learned methods with no novel insight or complex problem-solving. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Graph with correct shape: two parts not joined, values at 6/25, 12/25 shown | M1 | Two parts must be right shape and not joined; ignore labels; condone if goes below \(x\)-axis |
| Correct values at 6/25, 12/25, with domain 1, 2 and 4 shown, not going beyond 4 or < 1 | A1 | Can allow "freehand" straight line |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{d\left(\frac{3}{50}(4y^2 - y^3)\right)}{dy} = \frac{3}{50}(8y - 3y^2)\) | M1 | 1st M1: attempting to differentiate \(y^n \to y^{n-1}\) for \(n = 2\) or 3 |
| \(\frac{3}{50}(8y - 3y^2) = 0\) | M1 | 2nd M1: equating differential \((\neq f(y))\) to zero and attempting to solve |
| \(y = \frac{8}{3}\) oe | A1 | For \(\frac{8}{3}\) oe; allow awrt 2.67; if \(y=0\) seen it must be rejected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(E(Y^2) = \int_1^2 \left(\frac{6}{25}y^3 - \frac{6}{25}y^2\right)dy + \int_2^4 \left(\frac{12}{50}y^4 - \frac{3}{50}y^5\right)dy\) | M1 | 1st M1: using \(\int y^2 f(y)\) for both parts and attempting integration |
| \(= \left[\frac{6}{100}y^4 - \frac{6}{75}y^3\right]_1^2 + \left[\frac{12}{250}y^5 - \frac{3}{300}y^6\right]_2^4\) | A1 | 1st A1: correct integration for both parts; ignore limits |
| \(= \left[\left(\frac{8}{25}\right) - \left(-\frac{1}{50}\right)\right] + \left[\left(\frac{1024}{125}\right) - \left(\frac{112}{125}\right)\right] = \frac{1909}{250}\) or 7.636 or 7.64 | dM1; A1 | 2nd dM1: dep on 1st M1, adding the 2 parts and substituting correct limits; 2nd A1: allow 7.64 or 7.636 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\text{Var}(Y) = \frac{``1909"}{250} - 2.696^2\) | M1 | M1: for "their part(c)" \(- 2.696^2\) |
| \(= 0.367584\) | A1 | awrt 0.368 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{2}(y-1) \times \frac{6}{25}(y-1) = 0.1\) or \(\int_1^x \frac{6}{25}(y-1)\,dy = 0.1\) | M1 | 1st M1: allow \(\frac{1}{2}t \times \frac{6}{25}(t-1) = 0.1\) or \(\int_1^x \frac{6}{25}(y-1)\,dy = 0.1\) and some integration and sub of 1 and \(x\) |
| \(\frac{1}{2}(y-1) \times \frac{6}{25}(y-1) = 0.1\) or \(\frac{6}{25}\left[\left(\frac{x^2}{2} - x\right) + \frac{1}{2}\right] = 0.1\) or \(\frac{6}{50}(x-1)^2 = 0.1\) | A1 | 1st A1: correct equation in any form |
| \((y-1)^2 = \frac{5}{6}\) or \(y = 1 \pm \sqrt{\frac{5}{6}}\); \(y = 1.9128...\) | dM1; A1 | 2nd dM1: dep on 1st M1, correct method for solving; 2nd A1: awrt 1.91 (second solution rejected) |
# Question 3:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Graph with correct shape: two parts not joined, values at 6/25, 12/25 shown | M1 | Two parts must be right shape and not joined; ignore labels; condone if goes below $x$-axis |
| Correct values at 6/25, 12/25, with domain 1, 2 and 4 shown, not going beyond 4 or < 1 | A1 | Can allow "freehand" straight line |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{d\left(\frac{3}{50}(4y^2 - y^3)\right)}{dy} = \frac{3}{50}(8y - 3y^2)$ | M1 | 1st M1: attempting to differentiate $y^n \to y^{n-1}$ for $n = 2$ or 3 |
| $\frac{3}{50}(8y - 3y^2) = 0$ | M1 | 2nd M1: equating differential $(\neq f(y))$ to zero and attempting to solve |
| $y = \frac{8}{3}$ oe | A1 | For $\frac{8}{3}$ oe; allow awrt 2.67; if $y=0$ seen it must be rejected |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(Y^2) = \int_1^2 \left(\frac{6}{25}y^3 - \frac{6}{25}y^2\right)dy + \int_2^4 \left(\frac{12}{50}y^4 - \frac{3}{50}y^5\right)dy$ | M1 | 1st M1: using $\int y^2 f(y)$ for both parts and attempting integration |
| $= \left[\frac{6}{100}y^4 - \frac{6}{75}y^3\right]_1^2 + \left[\frac{12}{250}y^5 - \frac{3}{300}y^6\right]_2^4$ | A1 | 1st A1: correct integration for both parts; ignore limits |
| $= \left[\left(\frac{8}{25}\right) - \left(-\frac{1}{50}\right)\right] + \left[\left(\frac{1024}{125}\right) - \left(\frac{112}{125}\right)\right] = \frac{1909}{250}$ or **7.636** or **7.64** | dM1; A1 | 2nd dM1: dep on 1st M1, adding the 2 parts and substituting correct limits; 2nd A1: allow 7.64 or 7.636 |
## Part (d)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\text{Var}(Y) = \frac{``1909"}{250} - 2.696^2$ | M1 | M1: for "their part(c)" $- 2.696^2$ |
| $= 0.367584$ | A1 | awrt **0.368** |
## Part (e)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2}(y-1) \times \frac{6}{25}(y-1) = 0.1$ or $\int_1^x \frac{6}{25}(y-1)\,dy = 0.1$ | M1 | 1st M1: allow $\frac{1}{2}t \times \frac{6}{25}(t-1) = 0.1$ or $\int_1^x \frac{6}{25}(y-1)\,dy = 0.1$ and some integration and sub of 1 and $x$ |
| $\frac{1}{2}(y-1) \times \frac{6}{25}(y-1) = 0.1$ or $\frac{6}{25}\left[\left(\frac{x^2}{2} - x\right) + \frac{1}{2}\right] = 0.1$ or $\frac{6}{50}(x-1)^2 = 0.1$ | A1 | 1st A1: correct equation in any form |
| $(y-1)^2 = \frac{5}{6}$ or $y = 1 \pm \sqrt{\frac{5}{6}}$; $y = 1.9128...$ | dM1; A1 | 2nd dM1: dep on 1st M1, correct method for solving; 2nd A1: awrt **1.91** (second solution rejected) |
---\begin{enumerate}
\item The continuous random variable $Y$ has the following probability density function
\end{enumerate}
$$f ( y ) = \begin{cases} \frac { 6 } { 25 } ( y - 1 ) & 1 \leqslant y < 2 \\ \frac { 3 } { 50 } \left( 4 y ^ { 2 } - y ^ { 3 } \right) & 2 \leqslant y < 4 \\ 0 & \text { otherwise } \end{cases}$$
(a) Sketch f(y)\\
(b) Find the mode of $Y$\\
(c) Use algebraic integration to calculate $\mathrm { E } \left( Y ^ { 2 } \right)$
Given that $\mathrm { E } ( Y ) = 2.696$\\
(d) find $\operatorname { Var } ( Y )$\\
(e) Find the value of $y$ for which $\mathrm { P } ( Y \geqslant y ) = 0.9$
Give your answer to 3 significant figures.
\hfill \mbox{\textit{Edexcel S2 2021 Q3 [15]}}