| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2007 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Estimator properties and bias |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard probability distributions. Parts (i)-(iv) involve routine integration and algebraic manipulation (finding k, expectation, variance, CDF, median). Part (v) applies the Central Limit Theorem directly with given variance. While it requires multiple techniques, each step follows standard procedures without requiring novel insight or complex problem-solving, making it slightly easier than average. |
| Spec | 5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03f Relate pdf-cdf: medians and percentiles5.05a Sample mean distribution: central limit theorem |
1 The continuous random variable $X$ has probability density function
$$f ( x ) = k ( 1 - x ) \quad \text { for } 0 \leqslant x \leqslant 1$$
where $k$ is a constant.\\
(i) Show that $k = 2$. Sketch the graph of the probability density function.\\
(ii) Find $\mathrm { E } ( X )$ and show that $\operatorname { Var } ( X ) = \frac { 1 } { 18 }$.\\
(iii) Derive the cumulative distribution function of $X$. Hence find the probability that $X$ is greater than the mean.\\
(iv) Verify that the median of $X$ is $1 - \frac { 1 } { \sqrt { 2 } }$.\\
(v) $\bar { X }$ is the mean of a random sample of 100 observations of $X$. Write down the approximate distribution of $\bar { X }$.
\hfill \mbox{\textit{OCR MEI S3 2007 Q1 [18]}}