| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | October |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Find cumulative distribution F(x) |
| Difficulty | Standard +0.3 This is a multi-part question on discrete probability distributions requiring understanding of symmetry, cumulative distribution, and variance calculations. Part (a) uses symmetry (trivial), part (b) is straightforward algebra with F(0), part (c) requires setting up and solving simultaneous equations using variance formula, and part (d) applies Pythagoras theorem. While it has multiple steps and requires careful algebraic manipulation, all techniques are standard S1 content with no novel insight required, making it slightly easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| \(y\) | - 9 | - 5 | 0 | 5 | 9 |
| \(\mathrm { P } ( Y = y )\) | \(q\) | \(r\) | \(u\) | \(r\) | \(q\) |
| Answer | Marks | Guidance |
|---|---|---|
| By symmetry \(E(Y) = \mathbf{0}\) | B1 | For 0 |
| Answer | Marks | Guidance |
|---|---|---|
| \(q + r + u = \frac{19}{30}\) | M1 | Correct equation in \(q\), \(r\) and \(u\) using F(0) |
| \(2(q + r) + u = 1\) [and attempt to solve] | M1 | Second equation based on sum of probs = 1 and attempt to solve |
| \(u = \frac{8}{30} = \frac{4}{15}\) | A1*cso | Correct value for \(u\) with no incorrect working |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(Y^2) = (-9)^2 \times q + (-5)^2 \times r + 5^2 \times r + 9^2 q\) or \(162q + 50r\) | M1 | Attempt at \(E(Y^2)\) with at least 3 correct products |
| \(\text{Var}(Y) = 37 = E(Y^2) - 0^2 \Rightarrow 37 = 162q + 50r\) | dM1 | Attempt at correct equation in \(q\) and \(r\) using Var\((Y)\) |
| Solving with \(q + r = \frac{11}{30}\), e.g. \((162 - 50)q = 37 - \frac{55}{3}\) | M1 | Using \(q + r = \frac{11}{30}\) to solve two linear equations |
| \(q = \dfrac{1}{6}\) and \(r = \dfrac{1}{5}\) | A1 | For \(q = \frac{1}{6}\) and \(r = \frac{1}{5}\) or exact equivalents |
| Answer | Marks | Guidance |
|---|---|---|
| \(Y = 0 \Rightarrow D = 12\); \(D = \sqrt{12^2 + Y^2}\); \(Y = \pm 5 \Rightarrow D = 13\) or \(Y = \pm 9 \Rightarrow D = 15\) | B1, M1, A1 | For \(D = 12\); use of Pythagoras for \(D = 13\) or 15; \(D = 13\) and 15 |
| \(d\): 12, 13, 15 | M1A1ftA1ft | Correct values with associated probabilities ft their \(q\) and \(r\) |
| \(P(D=d)\): \(\frac{4}{15}\), \(\frac{6}{15}\) or \(\frac{2}{5}\), \(\frac{5}{15}\) or \(\frac{1}{3}\) | Fully correct probability distribution |
# Question 5:
## Part (a)
| By symmetry $E(Y) = \mathbf{0}$ | B1 | For 0 |
## Part (b)
| $q + r + u = \frac{19}{30}$ | M1 | Correct equation in $q$, $r$ and $u$ using F(0) |
| $2(q + r) + u = 1$ [and attempt to solve] | M1 | Second equation based on sum of probs = 1 and attempt to solve |
| $u = \frac{8}{30} = \frac{4}{15}$ | A1*cso | Correct value for $u$ with no incorrect working |
## Part (c)
| $E(Y^2) = (-9)^2 \times q + (-5)^2 \times r + 5^2 \times r + 9^2 q$ or $162q + 50r$ | M1 | Attempt at $E(Y^2)$ with at least 3 correct products |
| $\text{Var}(Y) = 37 = E(Y^2) - 0^2 \Rightarrow 37 = 162q + 50r$ | dM1 | Attempt at correct equation in $q$ and $r$ using Var$(Y)$ |
| Solving with $q + r = \frac{11}{30}$, e.g. $(162 - 50)q = 37 - \frac{55}{3}$ | M1 | Using $q + r = \frac{11}{30}$ to solve two linear equations |
| $q = \dfrac{1}{6}$ **and** $r = \dfrac{1}{5}$ | A1 | For $q = \frac{1}{6}$ and $r = \frac{1}{5}$ or exact equivalents |
## Part (d)
| $Y = 0 \Rightarrow D = 12$; $D = \sqrt{12^2 + Y^2}$; $Y = \pm 5 \Rightarrow D = 13$ or $Y = \pm 9 \Rightarrow D = 15$ | B1, M1, A1 | For $D = 12$; use of Pythagoras for $D = 13$ or 15; $D = 13$ and 15 |
| $d$: 12, 13, 15 | M1A1ftA1ft | Correct values with associated probabilities ft their $q$ and $r$ |
| $P(D=d)$: $\frac{4}{15}$, $\frac{6}{15}$ or $\frac{2}{5}$, $\frac{5}{15}$ or $\frac{1}{3}$ | | Fully correct probability distribution |
---\begin{enumerate}
\item The discrete random variable $Y$ has the following probability distribution
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$y$ & - 9 & - 5 & 0 & 5 & 9 \\
\hline
$\mathrm { P } ( Y = y )$ & $q$ & $r$ & $u$ & $r$ & $q$ \\
\hline
\end{tabular}
\end{center}
where $q , r$ and $u$ are probabilities.\\
(a) Write down the value of $\mathrm { E } ( Y )$
The cumulative distribution function of $Y$ is $\mathrm { F } ( y )$\\
Given that $F ( 0 ) = \frac { 19 } { 30 }$\\
(b) show that the value of $u$ is $\frac { 4 } { 15 }$
Given also that $\operatorname { Var } ( Y ) = 37$\\
(c) find the value of $q$ and the value of $r$
The coordinates of a point $P$ are $( 12 , Y )$
The random variable $D$ represents the length of $O P$\\
(d) Find the probability distribution of $D$
\hfill \mbox{\textit{Edexcel S1 2021 Q5 [14]}}