| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Expectation and variance from distribution |
| Difficulty | Standard +0.3 This is a straightforward Further Statistics question testing standard expectation/variance formulas and probability axioms. Parts (a)-(c) involve routine calculations with given distributions, while part (d) requires recognizing that P(W > T) = P(W > 3W - 8) simplifies to P(W < 4), which is a simple algebraic manipulation. All techniques are standard textbook exercises with no novel insight required, though the multi-part structure and Further Maths context place it slightly above average difficulty. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| \(x\) | - 3 | - 2 | - 1 | 0 | 2 | 5 |
| \(\mathrm { P } ( X = x )\) | 0.3 | 0.15 | 0.1 | 0.15 | 0.1 | 0.2 |
| \(y\) | - 2 | - 1 | 0 | 1 | 2 |
| \(\mathrm { P } ( Y = y )\) | \(3 a\) | \(a\) | \(b\) | \(a\) | \(c\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(X) = -0.1\) oe | B1 | \(-0.1\) oe |
| (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Var}(X) = E(X^2) - (\text{"-0.1"})^2\) | M1 | For recalling and using a correct formula |
| \(E(X^2) = 8.8\) | A1 | 8.8 |
| (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((-2)^2 \times 3a + (-1)^2 \times a[+0^2 \times b] + 1^2 \times a + 2^2 \times c = [\text{"2"}]\) | M1 | For use of \(\sum y^2 P(Y=y)[=2]\) or \(\sum(y^2+3)P(Y=y)[=5]\), 3 correct products seen |
| \(7a + 2c = 1\) oe | A1 | For correct equation with \(a\)'s collected |
| One of \(a+c=0.25\) or \(4a+b=0.75\) or \(5a+b+c=1\) | M1 | For use of \(\sum P(Y=y)=1\) or \(P(Y \leq 0)=0.75\) or \(1-P(Y \leq 0)=0.25\) |
| Two of \(a+c=0.25\) or \(4a+b=0.75\) or \(5a+b+c=1\) | A1 | For 2 correct equations |
| \(a=0.1\) and \(b=0.35\) and \(c=0.15\) | A1 | \(a\), \(b\) and \(c\) correct. Award full marks if all 3 correct |
| (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(W>T) = P(W > 3W-8) = P(W < 4)\) | M1 | For using the information given to work out the values of \(W\). Allow \(Y-X\) instead of \(W\) |
| \(P(W<4) = 1-[P(X=-3)\times P(Y=1) + P(X=-3)\times P(Y=2) + P(X=-2)\times P(Y=2)]\) or \(= P(X \geq -1) + P(X=-2)\times P(Y \neq 2) + P(X=-3)\times P(Y \leq 0)\) | M1dep | For using the information given to work out which are the relevant combinations of \(X\) and \(Y\). The irrelevant ones must not be used |
| \(= 1-[0.3\times\text{"0.1"}+0.3\times\text{"0.15"}+0.15\times\text{"0.15"}]\) or \(0.55 + 0.15\times[1-\text{"0.15"}]+0.3\times[\text{"0.3"}+\text{"0.1"}+\text{"0.35"}]\) | M1dep | Previous method must be awarded. All required cases identified and their probabilities of \(a\), \(b\) and \(c\) used. Allow in terms of \(a\), \(b\) and \(c\) |
| \(= \mathbf{0.9025}\) | A1 | 0.9025 (accept awrt 0.903 or exact fraction \(\frac{361}{400}\)) |
| (4) |
# Question 3:
## Part 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(X) = -0.1$ oe | B1 | $-0.1$ oe |
| | **(1)** | |
## Part 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Var}(X) = E(X^2) - (\text{"-0.1"})^2$ | M1 | For recalling and using a correct formula |
| $E(X^2) = 8.8$ | A1 | 8.8 |
| | **(2)** | |
## Part 3(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(-2)^2 \times 3a + (-1)^2 \times a[+0^2 \times b] + 1^2 \times a + 2^2 \times c = [\text{"2"}]$ | M1 | For use of $\sum y^2 P(Y=y)[=2]$ or $\sum(y^2+3)P(Y=y)[=5]$, 3 correct products seen |
| $7a + 2c = 1$ oe | A1 | For correct equation with $a$'s collected |
| One of $a+c=0.25$ **or** $4a+b=0.75$ **or** $5a+b+c=1$ | M1 | For use of $\sum P(Y=y)=1$ or $P(Y \leq 0)=0.75$ or $1-P(Y \leq 0)=0.25$ |
| Two of $a+c=0.25$ **or** $4a+b=0.75$ **or** $5a+b+c=1$ | A1 | For 2 correct equations |
| $a=0.1$ and $b=0.35$ and $c=0.15$ | A1 | $a$, $b$ and $c$ correct. Award full marks if all 3 correct |
| | **(5)** | |
## Part 3(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(W>T) = P(W > 3W-8) = P(W < 4)$ | M1 | For using the information given to work out the values of $W$. Allow $Y-X$ instead of $W$ |
| $P(W<4) = 1-[P(X=-3)\times P(Y=1) + P(X=-3)\times P(Y=2) + P(X=-2)\times P(Y=2)]$ **or** $= P(X \geq -1) + P(X=-2)\times P(Y \neq 2) + P(X=-3)\times P(Y \leq 0)$ | M1dep | For using the information given to work out which are the relevant combinations of $X$ and $Y$. The irrelevant ones must not be used |
| $= 1-[0.3\times\text{"0.1"}+0.3\times\text{"0.15"}+0.15\times\text{"0.15"}]$ **or** $0.55 + 0.15\times[1-\text{"0.15"}]+0.3\times[\text{"0.3"}+\text{"0.1"}+\text{"0.35"}]$ | M1dep | Previous method must be awarded. All required cases identified and their probabilities of $a$, $b$ and $c$ used. Allow in terms of $a$, $b$ and $c$ |
| $= \mathbf{0.9025}$ | A1 | 0.9025 (accept awrt 0.903 or exact fraction $\frac{361}{400}$) |
| | **(4)** | |
---\begin{enumerate}
\item The discrete random variable $X$ has probability distribution
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$x$ & - 3 & - 2 & - 1 & 0 & 2 & 5 \\
\hline
$\mathrm { P } ( X = x )$ & 0.3 & 0.15 & 0.1 & 0.15 & 0.1 & 0.2 \\
\hline
\end{tabular}
\end{center}
(a) Find $\mathrm { E } ( X )$
Given that $\operatorname { Var } ( X ) = 8.79$\\
(b) find $\mathrm { E } \left( X ^ { 2 } \right)$
The discrete random variable $Y$ has probability distribution
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$y$ & - 2 & - 1 & 0 & 1 & 2 \\
\hline
$\mathrm { P } ( Y = y )$ & $3 a$ & $a$ & $b$ & $a$ & $c$ \\
\hline
\end{tabular}
\end{center}
where $a$, $b$ and $c$ are constants.\\
For the random variable $Y$
$$\mathrm { P } ( Y \leqslant 0 ) = 0.75 \quad \text { and } \quad \mathrm { E } \left( Y ^ { 2 } + 3 \right) = 5$$
(c) Find the value of $a$, the value of $b$ and the value of $c$
The random variable $W = Y - X$ where $Y$ and $X$ are independent.\\
The random variable $T = 3 W - 8$\\
(d) Calculate $\mathrm { P } ( W > T )$
\hfill \mbox{\textit{Edexcel FS1 AS 2021 Q3 [12]}}