| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2017 |
| Session | December |
| Marks | 10 |
| Topic | Uniform Distribution |
| Type | Derive general variance formula |
| Difficulty | Standard +0.3 This is a structured multi-part question on discrete uniform distribution that guides students through a variance derivation using given formulas, then applies standard properties of expectation and variance. Part (i) is algebraic manipulation with provided results, parts (ii)-(iii) are routine calculations, and part (iv) uses standard linear transformation properties (Var(aX+b) = a²Var(X)). While it requires careful algebra and understanding of distribution properties, it's more straightforward than average A-level questions since key formulas are provided and the path is clearly signposted. |
| Spec | 5.02e Discrete uniform distribution5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(aE(X) + b = 10, a^2 \text{ Var}(X) = 117\) | M1 | Both equations seen; Or exact equivalent |
| \(\text{Var}(X) = 52\) | B1ft | FT on their \(n\) |
| \(\Rightarrow a = \sqrt{117/52}\) | M1 | Solve to find one letter |
| \(= \frac{3}{2}\) or 1.5 and \(b = -9.5\) | A1 | Both, exact or correct to 3 s.f., www |
## (iv)
$aE(X) + b = 10, a^2 \text{ Var}(X) = 117$ | M1 | Both equations seen; Or exact equivalent
$\text{Var}(X) = 52$ | B1ft | FT on their $n$
$\Rightarrow a = \sqrt{117/52}$ | M1 | Solve to find one letter
$= \frac{3}{2}$ or 1.5 and $b = -9.5$ | A1 | Both, exact or correct to 3 s.f., www
---4 The discrete random variable $X$ has the distribution $\mathrm { U } ( n )$.\\
(i) Use the results $\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$ and $\mathrm { E } ( X ) = \frac { n + 1 } { 2 }$ to show that $\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)$. It is given that $\mathrm { E } ( X ) = 13$.\\
(ii) Find the value of $n$.\\
(iii) Find $\mathrm { P } ( X < 7.5 )$.
It is given that $\mathrm { E } ( a X + b ) = 10$ and $\operatorname { Var } ( a X + b ) = 117$, where $a$ and $b$ are positive.\\
(iv) Calculate the value of $a$ and the value of $b$.
\hfill \mbox{\textit{OCR FS1 AS 2017 Q4 [10]}}