| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2003 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Probabilities in table form with k |
| Difficulty | Easy -1.3 This is a straightforward textbook exercise testing basic probability distribution properties: summing probabilities to 1, calculating expectation and variance using standard formulas, and a simple probability calculation. All steps are routine recall with no problem-solving or insight required, making it easier than average. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| \(x\) | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = x )\) | \(3 c\) | \(4 c\) | \(5 c\) | \(6 c\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(c = 1/18 = 0.0556\) | M1 | For \(\sum p_i = 1\) |
| A1 | For correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Var\((X) = 1.17\) (\(= 95/81\))(= 160c - 2500c^2\() | M1 | Using correct formula for \)E(X)$ |
| A1ft | For correct expectation, ft on their c | |
| M1 | For correct variance formula | |
| A1ft | For correct answer ft on their c |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) \(P(X > 2.78) = 11c = 0.611\) (\(= 11/18\)) | M1 | For using their correct number of discrete values of \(X\) |
| A1 | For correct answer |
**(i)** $18c = 1$
$c = 1/18 = 0.0556$ | M1 | For $\sum p_i = 1$
A1 | For correct answer
**2 marks total**
**(ii)** $E(X) = 2.78$ ($= 25/9$)(= 50c)
Var$(X) = 1.17$ ($= 95/81$)(= 160c - 2500c^2$) | M1 | Using correct formula for $E(X)$
A1ft | For correct expectation, ft on their c
M1 | For correct variance formula
A1ft | For correct answer ft on their c
**4 marks total**
**(iii)** $P(X > 2.78) = 11c = 0.611$ ($= 11/18$) | M1 | For using their correct number of discrete values of $X$
A1 | For correct answer
**2 marks total**8 A discrete random variable $X$ has the following probability distribution.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 \\
\hline
$\mathrm { P } ( X = x )$ & $3 c$ & $4 c$ & $5 c$ & $6 c$ \\
\hline
\end{tabular}
\end{center}
(i) Find the value of the constant $c$.\\
(ii) Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.\\
(iii) Find $\mathrm { P } ( X > \mathrm { E } ( X ) )$.
\hfill \mbox{\textit{CAIE S1 2003 Q8 [8]}}