| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Verify algebraic PDF formula |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard PDF verification and properties. Part (i) requires routine integration of sin(πx), part (ii) uses symmetry to find E(X)=0.5, part (iii) involves solving a simple integral equation, and parts (iv)-(v) are bookwork. All techniques are standard for this module with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_0^1 \frac{\pi}{2}\sin(\pi x)\,dx = \left[-\frac{1}{2}\cos(\pi x)\right]_0^1 = \frac{1}{2} - \left(-\frac{1}{2}\right) = 1\) | M1 | Attempt to integrate \(f(x)\), limits \((0,1)\) somewhere, evidence e.g. "from calculator" |
| B1 | Correctly integrate \(\sin(\pi x)\) to \(-\frac{1}{2}\cos(\pi x)\) | |
| A1 | Fully correct, need to see \(-\frac{1}{2}\cos(\pi x)\) and final 1, no wrong working seen | |
| and function non-negative for all \(x\) in range | B1 | Non-negative asserted explicitly; allow positive or equivalent. Not just graph drawn. *(Most will not get this mark!)* |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| [Graph: correct sine-curve shape through 0, above axis on \([0,1]\)] | M1 | Correct shape through 0; allow below axis outside range. Allow partial curve if clearly part of sine curve. |
| A1 | Fully correct including no extension beyond \([0,1]\). Don't worry about grads at ends. Ignore labelling of axes. | |
| \(E(X) = \frac{1}{2}\) | B1 | \(\frac{1}{2}\) or 0.5, needs to be simplified, no working needed, *no* ft |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_q^1 \frac{1}{2}\pi\sin(\pi x)\,dx = 0.75\); \(\;\left[-\frac{1}{2}\cos(\pi x)\right]_q^1 = 0.75\) | M1 | Equate integral to correct probability, correct limits *somewhere*; allow complementary probability (\(= 0.25\)) only if limits \((0, q)\) |
| \(\cos(\pi q) = 0.5\) | A1 | \(\cos(\pi q) = 0.5\) or exact equivalent |
| Solve to get \(q = \frac{1}{3}\) | A1 | \(q = \frac{1}{3}\) or a.r.t. 0.333. SR: Numerical (no working needed): 0.333 B3, 0.33 B2 |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_0^1 \frac{\pi}{2} x^2 \sin(\pi x)\,dx - \left(\frac{1}{2}\right)^2\) | M1 | Integral part correct; allow limits omitted, ignore \(dx\) |
| A1ft | Subtract their \([E(X)]^2\); allow \(\mu\) in form of integral, correct limits needed, not just "\(\mu^2\)" {note for scoris zoning – (ii) needs to be visible here} | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Values of \(x\) in range close to \(E(X)\) are more likely than those further away | B1 | Need to see "values of \(x\)" or equivalent, and probably not "occur". *Not* "the probability of \(x\) is greater when \(x\) is close to \(E(X)\)" etc. *Not* "PDF greater …" |
| [1] |
## Question 5:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_0^1 \frac{\pi}{2}\sin(\pi x)\,dx = \left[-\frac{1}{2}\cos(\pi x)\right]_0^1 = \frac{1}{2} - \left(-\frac{1}{2}\right) = 1$ | M1 | Attempt to integrate $f(x)$, limits $(0,1)$ somewhere, evidence e.g. "from calculator" |
| | B1 | Correctly integrate $\sin(\pi x)$ to $-\frac{1}{2}\cos(\pi x)$ |
| | A1 | Fully correct, need to see $-\frac{1}{2}\cos(\pi x)$ and final 1, no wrong working seen |
| and function non-negative for all $x$ in range | B1 | Non-negative asserted explicitly; allow positive or equivalent. Not just graph drawn. *(Most will not get this mark!)* |
| | **[4]** | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| [Graph: correct sine-curve shape through 0, above axis on $[0,1]$] | M1 | Correct shape through 0; allow below axis outside range. Allow partial curve if clearly part of sine curve. |
| | A1 | Fully correct including no extension beyond $[0,1]$. Don't worry about grads at ends. Ignore labelling of axes. |
| $E(X) = \frac{1}{2}$ | B1 | $\frac{1}{2}$ or 0.5, needs to be simplified, no working needed, *no* ft |
| | **[3]** | |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_q^1 \frac{1}{2}\pi\sin(\pi x)\,dx = 0.75$; $\;\left[-\frac{1}{2}\cos(\pi x)\right]_q^1 = 0.75$ | M1 | Equate integral to correct probability, correct limits *somewhere*; allow complementary probability ($= 0.25$) only if limits $(0, q)$ |
| $\cos(\pi q) = 0.5$ | A1 | $\cos(\pi q) = 0.5$ or exact equivalent |
| Solve to get $q = \frac{1}{3}$ | A1 | $q = \frac{1}{3}$ or a.r.t. 0.333. SR: Numerical (no working needed): 0.333 B3, 0.33 B2 |
| | **[3]** | |
### Part (iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_0^1 \frac{\pi}{2} x^2 \sin(\pi x)\,dx - \left(\frac{1}{2}\right)^2$ | M1 | Integral part correct; allow limits omitted, ignore $dx$ |
| | A1ft | Subtract their $[E(X)]^2$; allow $\mu$ in form of integral, correct limits needed, not just "$\mu^2$" {note for scoris zoning – (ii) needs to be visible here} |
| | **[2]** | |
### Part (v):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Values of $x$ in range close to $E(X)$ are more likely than those further away | B1 | Need to see "values of $x$" or equivalent, and probably not "occur". *Not* "the probability of $x$ is greater when $x$ is close to $E(X)$" etc. *Not* "PDF greater …" |
| | **[1]** | |
---5 A continuous random variable $X$ has probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 2 } \pi \sin ( \pi x ) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
(i) Show that this is a valid probability density function. [4]\\
(ii) Sketch the curve $\boldsymbol { y } = \mathbf { f } ( \boldsymbol { x } )$ and write down the value of $\mathbf { E } \boldsymbol { ( } \boldsymbol { X } \boldsymbol { ) }$. [3]\\
(iii) Find the value $q$ such that $\mathrm { P } ( X > q ) = 0.75$. [3]\\
(iv) Write down an expression, including an integral, for $\operatorname { Var } ( X )$. (Do not attempt to evaluate the integral.) [2]\\
(v) A student states that " $X$ is more likely to occur when $x$ is close to $\mathrm { E } ( X )$." Give an improved version of this statement. [1]
\hfill \mbox{\textit{OCR S2 2014 Q5 [13]}}