| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Discrete CDF to PMF |
| Difficulty | Moderate -0.8 This is a straightforward S1 question testing basic understanding of the relationship between PMF and CDF. Parts (a)-(c) require simple recall that F(x) is cumulative and sums to 1, involving only arithmetic. Parts (d)-(e) use basic independence and conditional probability with given probabilities. All steps are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 2.03a Mutually exclusive and independent events2.04a Discrete probability distributions |
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { p } ( x )\) | 0.10 | \(a\) | 0.28 | \(c\) | 0.24 |
| \(\mathrm {~F} ( x )\) | 0.10 | 0.26 | \(b\) | 0.76 | \(d\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((d =) 1\) | B1 | B1 for sight of 1 referring to \(d\) (may be in table or in the question) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a = 0.26 - 0.1\) or \(b = 0.26 + 0.28\) or \('a' + 0.38\) or \(0.76 - 'c'\) or \(c = 0.76 - 'b'\) or \(1 - (0.62 + 'a')\) | M1 | For any correct calculation seen (may be implied by one correct answer). ft their values for \(a\), \(b\) or \(c\). Do not award if answer is \(< 0\) or \(> 1\) |
| \(a = \mathbf{0.16}\), \(b = \mathbf{0.54}\), \(c = \mathbf{0.22}\) | A2 | A1 for at least two values correct; A2 for all 3 values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.24\) (only) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X \text{ is an odd number}) = 0.1 + 0.28 + 0.24 = 0.62\); \(P(X_1 \text{ and } X_2 \text{ are both odd}) = 0.62^2 = 0.3844\) | M1 | M1 for \((0.1 + 0.28 + 0.24)^2\) oe i.e. must be a complete correct expression, e.g. \((1 - ['a' + 'c'])^2\) and ft their values for \(a\) and \(c\) |
| \(= 0.3844\) awrt 0.384 | A1 | A1 for awrt 0.384 or exact fraction \(\frac{961}{2500}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X_1 + X_2 = 6 \mid \text{both are odd}) = \frac{P(X_1 + X_2 = 6 \cap X_1 \text{ and } X_2 \text{ are odd})}{P(X_1 \text{ and } X_2 \text{ are odd})}\) | M1 | For attempt at correct conditional probability i.e. a correct ratio of probabilities stated in words that mentions both \(X_1\) and \(X_2\). May be implied by a numerical ratio with correct numerator and their "(d)" on denominator. This would score M1A1ft |
| \(= \frac{0.1 \times 0.24 + 0.28 \times 0.28 + 0.24 \times 0.1}{\text{(their answer to d)}} = \frac{0.1264}{\text{(d)}}\) | A1ft | 1st A1ft for \(\frac{\text{correct numerator}}{0.384...}\) or correct numerator and denominator of their 'd' |
| \(= 0.3288...\) awrt 0.329 | A1 | 2nd A1 for awrt 0.329 or exact fraction \(\frac{316}{961}\) |
# Question 1:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(d =) 1$ | B1 | B1 for sight of 1 referring to $d$ (may be in table or in the question) |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $a = 0.26 - 0.1$ or $b = 0.26 + 0.28$ or $'a' + 0.38$ or $0.76 - 'c'$ or $c = 0.76 - 'b'$ or $1 - (0.62 + 'a')$ | M1 | For any correct calculation seen (may be implied by one correct answer). ft their values for $a$, $b$ or $c$. Do not award if answer is $< 0$ or $> 1$ |
| $a = \mathbf{0.16}$, $b = \mathbf{0.54}$, $c = \mathbf{0.22}$ | A2 | A1 for at least two values correct; A2 for all 3 values correct |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.24$ (only) | B1 | |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X \text{ is an odd number}) = 0.1 + 0.28 + 0.24 = 0.62$; $P(X_1 \text{ and } X_2 \text{ are both odd}) = 0.62^2 = 0.3844$ | M1 | M1 for $(0.1 + 0.28 + 0.24)^2$ oe i.e. must be a complete correct expression, e.g. $(1 - ['a' + 'c'])^2$ and ft their values for $a$ and $c$ |
| $= 0.3844$ **awrt 0.384** | A1 | A1 for awrt 0.384 or exact fraction $\frac{961}{2500}$ |
## Part (e)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X_1 + X_2 = 6 \mid \text{both are odd}) = \frac{P(X_1 + X_2 = 6 \cap X_1 \text{ and } X_2 \text{ are odd})}{P(X_1 \text{ and } X_2 \text{ are odd})}$ | M1 | For attempt at correct conditional probability i.e. a correct ratio of probabilities stated in words that mentions both $X_1$ and $X_2$. May be implied by a numerical ratio with correct numerator and their "(d)" on denominator. This would score M1A1ft |
| $= \frac{0.1 \times 0.24 + 0.28 \times 0.28 + 0.24 \times 0.1}{\text{(their answer to d)}} = \frac{0.1264}{\text{(d)}}$ | A1ft | 1st A1ft for $\frac{\text{correct numerator}}{0.384...}$ or correct numerator and denominator of their 'd' |
| $= 0.3288...$ **awrt 0.329** | A1 | 2nd A1 for awrt 0.329 or exact fraction $\frac{316}{961}$ |
---\begin{enumerate}
\item The discrete random variable $X$ has probability function $\mathrm { p } ( x )$ and cumulative distribution function $\mathrm { F } ( x )$ given in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$\mathrm { p } ( x )$ & 0.10 & $a$ & 0.28 & $c$ & 0.24 \\
\hline
$\mathrm {~F} ( x )$ & 0.10 & 0.26 & $b$ & 0.76 & $d$ \\
\hline
\end{tabular}
\end{center}
(a) Write down the value of $d$\\
(b) Find the values of $a$, $b$ and $c$\\
(c) Write down the value of $\mathrm { P } ( X > 4 )$
Two independent observations, $X _ { 1 }$ and $X _ { 2 }$, are taken from the distribution of $X$.\\
(d) Find the probability that $X _ { 1 }$ and $X _ { 2 }$ are both odd.
Given that $X _ { 1 }$ and $X _ { 2 }$ are both odd,\\
(e) find the probability that the sum of $X _ { 1 }$ and $X _ { 2 }$ is 6
Give your answer to 3 significant figures.\\
\hfill \mbox{\textit{Edexcel S1 2015 Q1 [10]}}