| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Single-piece PDF with k |
| Difficulty | Standard +0.3 This is a standard S2 probability density function question requiring integration to find k, variance calculation, and solving an equation. All techniques are routine for this module, though part (iii) requires careful algebraic manipulation. Slightly above average due to the multi-step nature and algebraic complexity, but follows predictable patterns for S2 questions. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_1^\infty kx^{-a}\,dx = \left[k\frac{x^{-a+1}}{-a+1}\right]_1^\infty\) | M1 | Integrate \(f(x)\), limits 1 and \(\infty\) (at some stage) |
| B1 | Correct indefinite integral | |
| Correctly obtain \(k = a-1\) AG | A1 3 | Correctly obtain given answer; treatment of \(\infty\) must not be wrong; not \(k^{-a+1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_1^\infty 3x^{-3}\,dx = \left[3\frac{x^{-2}}{-2}\right]_1^\infty = 1\tfrac{1}{2}\) | M1 | Integrate \(xf(x)\), limits 1 and \(\infty\); [\(x^3\) is *not* MR] |
| \(\int_1^\infty 3x^{-2}\,dx = \left[3\frac{x^{-1}}{-1}\right]_1^\infty - (1\tfrac{1}{2})^2\) | M1 | Integrate \(x^2f(x)\), correct limits |
| A1 | Either \(\mu = 1\tfrac{1}{2}\) or \(E(X^2)=3\) stated or implied, allow \(k\), \(k/2\) | |
| M1 | Subtract their numerical \(\mu^2\), allow letter if subs later | |
| Answer \(\frac{3}{4}\) | A1 5 | Final answer \(\frac{3}{4}\) or 0.75 only, cwo, e.g. not from \(\mu=-1\tfrac{1}{2}\). [SR: Limits 0,1: can get (i) B1, (ii) M1M1M1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int_1^2 (a-1)x^{-a}\,dx = \left[-x^{-a+1}\right]_1^2 = 0.9\) | M1* | Equate \(\int f(x)\,dx\), one limit 2, to 0.9 or 0.1. [Normal: 0 ex 4] |
| \(1 - \frac{1}{2^{a-1}} = 0.9,\quad 2^{a-1} = 10\) | dep\*M1 | Solve equation of this form to get \(2^{a-1} =\) number |
| M1 indep | Use logs or equivalent to solve \(2^{a-1} =\) number | |
| \(a = 4.322\) | A1 4 | Answer a.r.t. 4.32. T&I: (M1M1) B2 or B0 |
# Question 8:
## Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_1^\infty kx^{-a}\,dx = \left[k\frac{x^{-a+1}}{-a+1}\right]_1^\infty$ | M1 | Integrate $f(x)$, limits 1 and $\infty$ (at some stage) |
| | B1 | Correct indefinite integral |
| Correctly obtain $k = a-1$ **AG** | A1 **3** | Correctly obtain given answer; treatment of $\infty$ must not be wrong; not $k^{-a+1}$ |
## Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_1^\infty 3x^{-3}\,dx = \left[3\frac{x^{-2}}{-2}\right]_1^\infty = 1\tfrac{1}{2}$ | M1 | Integrate $xf(x)$, limits 1 and $\infty$; [$x^3$ is *not* MR] |
| $\int_1^\infty 3x^{-2}\,dx = \left[3\frac{x^{-1}}{-1}\right]_1^\infty - (1\tfrac{1}{2})^2$ | M1 | Integrate $x^2f(x)$, correct limits |
| | A1 | Either $\mu = 1\tfrac{1}{2}$ or $E(X^2)=3$ stated or implied, allow $k$, $k/2$ |
| | M1 | Subtract their numerical $\mu^2$, allow letter if subs later |
| Answer $\frac{3}{4}$ | A1 **5** | Final answer $\frac{3}{4}$ or 0.75 only, cwo, e.g. not from $\mu=-1\tfrac{1}{2}$. [SR: Limits 0,1: can get (i) B1, (ii) M1M1M1] |
## Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_1^2 (a-1)x^{-a}\,dx = \left[-x^{-a+1}\right]_1^2 = 0.9$ | M1* | Equate $\int f(x)\,dx$, one limit 2, to 0.9 or 0.1. [Normal: 0 ex 4] |
| $1 - \frac{1}{2^{a-1}} = 0.9,\quad 2^{a-1} = 10$ | dep\*M1 | Solve equation of this form to get $2^{a-1} =$ number |
| | M1 indep | Use logs or equivalent to solve $2^{a-1} =$ number |
| $a = 4.322$ | A1 **4** | Answer a.r.t. 4.32. T&I: (M1M1) B2 or B0 |
---8 The continuous random variable $X$ has probability density function given by
$$\mathrm { f } ( x ) = \begin{cases} k x ^ { - a } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
where $k$ and $a$ are constants and $a$ is greater than 1 .\\
(i) Show that $k = a - 1$.\\
(ii) Find the variance of $X$ in the case $a = 4$.\\
(iii) It is given that $\mathrm { P } ( X < 2 ) = 0.9$. Find the value of $a$, correct to 3 significant figures.
\hfill \mbox{\textit{OCR S2 2010 Q8 [12]}}