| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Functions of random variables |
| Difficulty | Standard +0.8 This Further Statistics question requires computing variance from a probability distribution, working with a piecewise-defined function of a random variable, and finding E(XY). While the individual calculations are methodical, part (c) requires constructing the joint distribution XY and careful bookkeeping across cases, which elevates it above routine A-level questions but doesn't require deep insight. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p) |
| \(x\) | - 5 | - 2 | 3 | 4 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 12 }\) | \(\frac { 1 } { 6 }\) | \(\frac { 1 } { 4 }\) | \(\frac { 1 } { 2 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([\text{E}(X) =] (-5)\times\frac{1}{12} + (-2)\times\frac{1}{6} + (3)\times\frac{1}{4} + (4)\times\frac{1}{2} [=2]\) | M1 | Attempt at \(\text{E}(X)\) with at least 3 correct products seen |
| \([\text{E}(X^2) =] (-5)^2\times\frac{1}{12} + (-2)^2\times\frac{1}{6} + (3)^2\times\frac{1}{4} + (4)^2\times\frac{1}{2} [=13]\) | M1 | Attempt at \(\text{E}(X^2)\) with at least 3 correct products seen |
| \(\text{Var}(X) = \text{E}(X^2) - [\text{E}(X)]^2 = 13 - 2^2 = \mathbf{9}\) | A1 | 9 cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Distribution of \(Y\): \(x\): \((-5), -2, 3, (4)\); \(y\): \((25), 4, 7, (10)\); \(p\): \((\frac{1}{12}), \frac{1}{6}, \frac{1}{4}, (\frac{1}{2})\) | M1 | Finding distribution of \(Y\) |
| \(P(Y < 9) = P(X=-2) + P(X=3) [= \frac{1}{6} + \frac{1}{4}]\) | M1 | \(P(X=-2)+P(X=3)\) or \(P(Y=4)+P(Y=7)\) |
| \(= \frac{5}{12}\) | A1 | Condone awrt 0.417 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{E}(XY) = (-5)(25)\frac{1}{12} + (-2)(4)\times\frac{1}{6} + (3)(7)\times\frac{1}{4} + (4)(10)\times\frac{1}{2}\) | M1 | Attempt at \(\text{E}(XY)\) with at least 2 correct terms |
| \(= 13.5\) | A1 |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[\text{E}(X) =] (-5)\times\frac{1}{12} + (-2)\times\frac{1}{6} + (3)\times\frac{1}{4} + (4)\times\frac{1}{2} [=2]$ | M1 | Attempt at $\text{E}(X)$ with at least 3 correct products seen |
| $[\text{E}(X^2) =] (-5)^2\times\frac{1}{12} + (-2)^2\times\frac{1}{6} + (3)^2\times\frac{1}{4} + (4)^2\times\frac{1}{2} [=13]$ | M1 | Attempt at $\text{E}(X^2)$ with at least 3 correct products seen |
| $\text{Var}(X) = \text{E}(X^2) - [\text{E}(X)]^2 = 13 - 2^2 = \mathbf{9}$ | A1 | 9 cao |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Distribution of $Y$: $x$: $(-5), -2, 3, (4)$; $y$: $(25), 4, 7, (10)$; $p$: $(\frac{1}{12}), \frac{1}{6}, \frac{1}{4}, (\frac{1}{2})$ | M1 | Finding distribution of $Y$ |
| $P(Y < 9) = P(X=-2) + P(X=3) [= \frac{1}{6} + \frac{1}{4}]$ | M1 | $P(X=-2)+P(X=3)$ or $P(Y=4)+P(Y=7)$ |
| $= \frac{5}{12}$ | A1 | Condone awrt 0.417 |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{E}(XY) = (-5)(25)\frac{1}{12} + (-2)(4)\times\frac{1}{6} + (3)(7)\times\frac{1}{4} + (4)(10)\times\frac{1}{2}$ | M1 | Attempt at $\text{E}(XY)$ with at least 2 correct terms |
| $= 13.5$ | A1 | |
---\begin{enumerate}
\item The discrete random variable $X$ has the following probability distribution.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & - 5 & - 2 & 3 & 4 \\
\hline
$\mathrm { P } ( X = x )$ & $\frac { 1 } { 12 }$ & $\frac { 1 } { 6 }$ & $\frac { 1 } { 4 }$ & $\frac { 1 } { 2 }$ \\
\hline
\end{tabular}
\end{center}
(a) Find $\operatorname { Var } ( X )$
The discrete random variable $Y$ is defined in terms of the discrete random variable $X$\\
When $X$ is negative, $Y = X ^ { 2 }$\\
When $X$ is positive, $Y = 3 X - 2$\\
(b) Find $\mathrm { P } ( Y < 9 )$\\
(c) Find $\mathrm { E } ( X Y )$
\hfill \mbox{\textit{Edexcel FS1 2020 Q4 [8]}}