| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Probability distributions with parameters |
| Difficulty | Standard +0.3 This is a straightforward Further Statistics 1 question requiring standard expectation/variance formulas and basic algebraic manipulation. Part (a) is routine calculation of E(X), part (b) involves solving a quadratic equation using Var(X) = E(X²) - [E(X)]², and part (c) requires solving a simple inequality. While it's Further Maths content, the techniques are mechanical with no novel insight required, making it slightly easier than average overall. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(x\) | - 5 | - 1 | 0 | 5 | \(b\) |
| \(\mathrm { P } ( X = x )\) | 0.3 | 0.25 | 0.1 | 0.15 | 0.2 |
| Answer | Marks | Guidance |
|---|---|---|
| \([E(X) =] 0.2b - 1\) | B1 | Correct expression for \(E(X)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X^2) = 25 \times 0.3 + 1 \times 0.25 [+0 \times 0.1] + 25 \times 0.15 + 0.2b^2 [= 11.5 + 0.2b^2]\) | M1 | Correct attempt at \(E(X^2)\) using \(\sum x^2 P(X=x)\), at least 3 correct non-zero products. Allow \((-5)^2\) etc |
| \("11.5 + 0.2b^2" - ("0.2b-1")^2 [= 34.26]\) | M1 | Realising that \(\text{Var}(X) = E(X^2) - [E(X)]^2\) needs to be used |
| \(0.16b^2 + 0.4b - 23.76 [=0]\) or \(\frac{4}{25}b^2 + \frac{2}{5}b - \frac{594}{25}[=0]\) | M1 | Reducing to a 3 term quadratic. At least 2 terms correct. Allow e.g. \(0.16b^2 + 0.4b = 23.76\). Condone missing "=0" |
| \(b = \mathbf{11}\) [since \(b > 5\)] | A1 | For 11 only (from correct equation), so \(-13.5\) must be eliminated. Correct answer with no incorrect working scores 4/4 |
| Answer | Marks | Guidance |
|---|---|---|
| At least 4 values correct for \((X^2\) and \(2-3X)\) or \((X^2-2\) and \(-3X)\) or \(X^2+3X\) or \(X^2+3X-2\) | M1 | Allow for solving equation with one sign error |
| All correct or correct ft with their \(b\), must have \(b>5\) (accurate to 1 sf) | A1ft | Allow solving equation to get awrt \(-3.6\) and awrt \(0.56\) or \(\frac{-3\pm\sqrt{17}}{2}\) (ft their \(b>5\)) |
| \(P(X^2 < 2-3X) = P(X=-1) + P(X=0)\) | M1 | For identifying correct values \(X=-1\) and \(X=0\) |
| \(= \mathbf{0.35}\) | A1 | NB: possible to score M0A0M1A1 if table incorrect. Correct answer with no incorrect working scores 4/4 |
## Question 2:
### Part (a):
| $[E(X) =] 0.2b - 1$ | B1 | Correct expression for $E(X)$ |
### Part (b):
| $E(X^2) = 25 \times 0.3 + 1 \times 0.25 [+0 \times 0.1] + 25 \times 0.15 + 0.2b^2 [= 11.5 + 0.2b^2]$ | M1 | Correct attempt at $E(X^2)$ using $\sum x^2 P(X=x)$, at least 3 correct non-zero products. Allow $(-5)^2$ etc |
| $"11.5 + 0.2b^2" - ("0.2b-1")^2 [= 34.26]$ | M1 | Realising that $\text{Var}(X) = E(X^2) - [E(X)]^2$ needs to be used |
| $0.16b^2 + 0.4b - 23.76 [=0]$ or $\frac{4}{25}b^2 + \frac{2}{5}b - \frac{594}{25}[=0]$ | M1 | Reducing to a 3 term quadratic. At least 2 terms correct. Allow e.g. $0.16b^2 + 0.4b = 23.76$. Condone missing "=0" |
| $b = \mathbf{11}$ [since $b > 5$] | A1 | For 11 only (from correct equation), so $-13.5$ must be eliminated. Correct answer with no incorrect working scores 4/4 |
### Part (c):
| At least 4 values correct for $(X^2$ and $2-3X)$ or $(X^2-2$ and $-3X)$ or $X^2+3X$ or $X^2+3X-2$ | M1 | Allow for solving equation with one sign error |
| All correct or correct ft with their $b$, must have $b>5$ (accurate to 1 sf) | A1ft | Allow solving equation to get awrt $-3.6$ and awrt $0.56$ or $\frac{-3\pm\sqrt{17}}{2}$ (ft their $b>5$) |
| $P(X^2 < 2-3X) = P(X=-1) + P(X=0)$ | M1 | For identifying correct values $X=-1$ and $X=0$ |
| $= \mathbf{0.35}$ | A1 | NB: possible to score M0A0M1A1 if table incorrect. Correct answer with no incorrect working scores 4/4 |
---\begin{enumerate}
\item The discrete random variable $X$ has probability distribution
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 5 & - 1 & 0 & 5 & $b$ \\
\hline
$\mathrm { P } ( X = x )$ & 0.3 & 0.25 & 0.1 & 0.15 & 0.2 \\
\hline
\end{tabular}
\end{center}
where $b$ is a constant and $b > 5$\\
(a) Find $\mathrm { E } ( X )$ in terms of $b$
Given that $\operatorname { Var } ( X ) = 34.26$\\
(b) find the value of $b$\\
(c) Find $\mathrm { P } \left( X ^ { 2 } < 2 - 3 X \right)$
\hfill \mbox{\textit{Edexcel FS1 2022 Q2 [9]}}