| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Find parameter from probability condition |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring integration of a given pdf and solving for a parameter. Part (i) involves basic integration and cube root extraction, part (ii) uses the variance formula E(X²) - [E(X)]² with symmetric distribution simplifying to E(X²) = Var(X), and part (iii) tests understanding of notation. All techniques are standard for S2 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 integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_{-3}^{3} \frac{3}{2a^3}x^2\,dx = \left[\frac{x^3}{2a^3}\right]_{-3}^{3} = \frac{27}{a^3}\) | M1dep* | Integrate, attempt at correct seen limits somewhere |
| Correct indefinite integral, can be implied e.g. \(27/a^3\) | B1 | Allow e.g. "\(< 3\)" \(=\) "\(\leq -4\)". Allow also for \(a^3\) on top |
| Equate, with limits, to 0.125 and solve | *M1 | |
| \(a = 6\) | A1 | Allow 6.00 but no other decimals. *Not* \(\pm 6\) |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mu = 0\) | B1 | Stated somewhere or calculated, any \(a\) |
| \(\int_{-a}^{a} kx^4\,dx = \left[k\frac{x^5}{5}\right]_{-a}^{a} = \frac{3a^2}{5}\) | M1dep* | Attempt to integrate \(x^2f(x)\), limits \(\pm a\) |
| Or exact equivalent, can be implied | B1 | |
| Equate to 1.35 and solve | *M1 | |
| \(a = 1.5 \pm 0.005\), allow \(\pm 1.5\), ignore "must be positive" | A1 | |
| \(a = \mathbf{1.5}\) | \(a = 3\) is *not* MR but can get B1 for \(\mu = 0\) | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\) is a value [values] that \(X\) takes | B1 | Ignore irrelevancies or extra wrong, unless contradictory |
| Total: 1 | *Not* answers just about the *function* |
## Question 3:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_{-3}^{3} \frac{3}{2a^3}x^2\,dx = \left[\frac{x^3}{2a^3}\right]_{-3}^{3} = \frac{27}{a^3}$ | M1dep* | Integrate, attempt at correct seen limits somewhere |
| Correct indefinite integral, can be implied e.g. $27/a^3$ | B1 | Allow e.g. "$< 3$" $=$ "$\leq -4$". Allow also for $a^3$ on top |
| Equate, with limits, to 0.125 and solve | *M1 | |
| $a = 6$ | A1 | Allow 6.00 but no other decimals. *Not* $\pm 6$ |
| **Total: 4** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mu = 0$ | B1 | Stated somewhere or calculated, any $a$ |
| $\int_{-a}^{a} kx^4\,dx = \left[k\frac{x^5}{5}\right]_{-a}^{a} = \frac{3a^2}{5}$ | M1dep* | Attempt to integrate $x^2f(x)$, limits $\pm a$ |
| Or exact equivalent, can be implied | B1 | |
| Equate to 1.35 and solve | *M1 | |
| $a = 1.5 \pm 0.005$, allow $\pm 1.5$, ignore "must be positive" | A1 | |
| $a = \mathbf{1.5}$ | | $a = 3$ is *not* MR but can get B1 for $\mu = 0$ |
| **Total: 5** | | |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x$ is a value [values] that $X$ takes | B1 | Ignore irrelevancies or extra wrong, unless contradictory |
| **Total: 1** | | *Not* answers just about the *function* |
---3 A continuous random variable $X$ has probability density function
$$f ( x ) = \left\{ \begin{array} { c l }
\frac { 3 } { 2 a ^ { 3 } } x ^ { 2 } & - a \leqslant x \leqslant a \\
0 & \text { otherwise }
\end{array} \right.$$
where $a$ is a constant.\\
(i) It is given that $\mathrm { P } ( - 3 \leqslant X \leqslant 3 ) = 0.125$. Find the value of $a$ in this case.\\
(ii) It is given instead that $\operatorname { Var } ( X ) = 1.35$. Find the value of $a$ in this case.\\
(iii) Explain the relationship between $x$ and $X$ in this question.
\hfill \mbox{\textit{OCR S2 2015 Q3 [10]}}