| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2003 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate Var(X) from probability function |
| Difficulty | Standard +0.3 This is a standard S1 probability distribution question requiring routine application of formulas for expectation, variance, and sum of independent random variables. While multi-part with 6 sections, each step follows textbook procedures: finding k from ΣP(X=x)=1, calculating E(X) and Var(X) using standard formulas, applying variance properties, and constructing a probability distribution for a sum. No novel insight or complex problem-solving required, making it slightly easier than average. |
| Spec | 2.04a Discrete probability distributions2.04b Binomial distribution: as model B(n,p)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2k + k + 0 + k = 1 \Rightarrow 4k = 1 \Rightarrow k = 0.25\) | M1 | |
| \(k = 0.25\) ✓ | A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{E}(X) = \sum x\text{P}(X=x) = 0 + 0.25 + 0 + 0.75 = 1\) | M1A1 | |
| \(\text{E}(X^2) = 0 + 0.25 + 0 + 2.25 = 2.5\) ✓ | M1A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var}(3X-2) = 3^2\text{Var}(X)\) | M1 | |
| \(= 9(2.5 - 1^2) = 13.5\) | M1A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{P}(X_1 + X_2) = \text{P}(X_1=3 \cap X_2=2) + \text{P}(X_1=2 \cap X_2=3) = 0+0=0\) | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(Y = X_1 + X_2\), values \(y\): 0, 1, 2, 3, 4, 5, 6 | B1 | |
| \(\text{P}(Y=y)\): 0.25, 0.25, 0.0625, 0.25, 0.125, (0), 0.0625 | B2 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{P}(1.3 \leq X_1 + X_2 \leq 3.2) = \text{P}(X_1+X_2=2) + \text{P}(X_1+X_2=3)\) | M1 | |
| \(= 0.0625 + 0.25 = 0.3125\) | A1ft, A1ft | (3 marks) |
# Question 5:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2k + k + 0 + k = 1 \Rightarrow 4k = 1 \Rightarrow k = 0.25$ | M1 | |
| $k = 0.25$ ✓ | A1 | **(2 marks)** |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{E}(X) = \sum x\text{P}(X=x) = 0 + 0.25 + 0 + 0.75 = 1$ | M1A1 | |
| $\text{E}(X^2) = 0 + 0.25 + 0 + 2.25 = 2.5$ ✓ | M1A1 | **(4 marks)** |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var}(3X-2) = 3^2\text{Var}(X)$ | M1 | |
| $= 9(2.5 - 1^2) = 13.5$ | M1A1 | **(3 marks)** |
## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{P}(X_1 + X_2) = \text{P}(X_1=3 \cap X_2=2) + \text{P}(X_1=2 \cap X_2=3) = 0+0=0$ | B1 | **(1 mark)** |
## Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $Y = X_1 + X_2$, values $y$: 0, 1, 2, 3, 4, 5, 6 | B1 | |
| $\text{P}(Y=y)$: 0.25, 0.25, 0.0625, 0.25, 0.125, (0), 0.0625 | B2 | **(3 marks)** |
## Part (f)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{P}(1.3 \leq X_1 + X_2 \leq 3.2) = \text{P}(X_1+X_2=2) + \text{P}(X_1+X_2=3)$ | M1 | |
| $= 0.0625 + 0.25 = 0.3125$ | A1ft, A1ft | **(3 marks)** |
---5. The discrete random variable $X$ has probability function
$$\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) , & x = 0,1,2 \\ k ( x - 2 ) , & x = 3 \\ 0 , & \text { otherwise } \end{cases}$$
where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = 0.25$.
\item Find $\mathrm { E } ( X )$ and show that $\mathrm { E } \left( X ^ { 2 } \right) = 2.5$.
\item Find $\operatorname { Var } ( 3 X - 2 )$.
Two independent observations $X _ { 1 }$ and $X _ { 2 }$ are made of $X$.
\item Show that $\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0$.
\item Find the complete probability function for $X _ { 1 } + X _ { 2 }$.
\item Find $\mathrm { P } \left( 1.3 \leq X _ { 1 } + X _ { 2 } \leq 3.2 \right)$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2003 Q5 [16]}}