| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.8 This is a straightforward S1 question requiring two standard techniques: using the sum of probabilities equals 1 and the definition of expectation to form simultaneous equations. The algebra is simple (linear equations), and part (ii) is routine calculation of variance/standard deviation using the standard formula. No problem-solving insight needed, just direct application of formulas. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 3 }\) | \(\frac { 1 } { 4 }\) | \(p\) | \(q\) |
| Answer | Marks | Guidance |
|---|---|---|
| 5(i) | \(^1/_{3} + ^1/_{4} + p + q = 1\) oe \(0 \times ^1/_{3} + 1 \times ^1/_{4} + 2p + 3q = 1\frac{1}{4}\) oe equalize coeffs, eg mult eqn (i) by 2 or 3 Or make \(p\) or \(q\) subject of (i) or (ii) | B1: —; B1: — |
| \(p = ^1/_{12}, q = ^1/_{6}\) oe | M1: —; A1A1: 5 | allow one error. If their equns subst or subtr not nec'y |
| 5(ii) | \(\Sigma x^2p\) (not \(i4\) or \(i3\) etc) \(= (2^1/_{4})\) \((1/_{10})^2\) | M1: —; M1: — |
| \(\text{ft (i) (0S } p, q < 1) \text{ or letters } p, q \text{ both M1s cao}\) | ||
| \(= 1.1875\) or \(1^7/_{16}\) oe sd \(=\sqrt{(\text{their } 1.1875)} = 1.09\) (3 sfs) | A1: —; B1f: 4 | dep 1st M1 &/(+ve no.) eg \(\sqrt{2.75 = 1.66}\) |
5(i) | $^1/_{3} + ^1/_{4} + p + q = 1$ oe $0 \times ^1/_{3} + 1 \times ^1/_{4} + 2p + 3q = 1\frac{1}{4}$ oe equalize coeffs, eg mult eqn (i) by 2 or 3 Or make $p$ or $q$ subject of (i) or (ii) | B1: —; B1: — | |
| $p = ^1/_{12}, q = ^1/_{6}$ oe | M1: —; A1A1: 5 | allow one error. If their equns subst or subtr not nec'y |
5(ii) | $\Sigma x^2p$ (not $i4$ or $i3$ etc) $= (2^1/_{4})$ $(1/_{10})^2$ | M1: —; M1: — | $\geq 2 \text{ non-zero terms correct, dep +ve result indep if +ve result or } , x-(1^r_{1/})/^p$ (≥ 2 (non-(0) terms correct): M2 |
| | | $\text{ft (i) (0S } p, q < 1) \text{ or letters } p, q \text{ both M1s cao}$ | |
| $= 1.1875$ or $1^7/_{16}$ oe sd $=\sqrt{(\text{their } 1.1875)} = 1.09$ (3 sfs) | A1: —; B1f: 4 | dep 1st M1 &/(+ve no.) eg $\sqrt{2.75 = 1.66}$ |
**Total: 9**
---5 The probability distribution of a discrete random variable, $X$, is given in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & $\frac { 1 } { 3 }$ & $\frac { 1 } { 4 }$ & $p$ & $q$ \\
\hline
\end{tabular}
\end{center}
It is given that the expectation, $\mathrm { E } ( X )$, is $1 \frac { 1 } { 4 }$.\\
(i) Calculate the values of $p$ and $q$.\\
(ii) Calculate the standard deviation of $X$.
\hfill \mbox{\textit{OCR S1 2006 Q5 [9]}}