| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2001 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | One unknown from sum constraint only |
| Difficulty | Easy -1.8 This is a routine S1 question testing basic probability distribution properties: finding a missing probability using the sum-to-1 constraint, calculating simple probabilities, evaluating the CDF, and applying standard formulae for E(aX+b) and Var(aX+b). All parts are direct applications of definitions with no problem-solving or insight required. |
| Spec | 2.04a Discrete probability distributions5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration |
| \(x\) | - 2 | - 1 | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | 0.1 | \(\alpha\) | 0.3 | 0.2 | 0.1 | 0.1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(w = \underline{0.2}\) | B1 (1) | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(-1 \leq X \leq 2) = P(0) + P(1) + P(2)\) | M1 | |
| \(= \underline{0.6}\) | A1 (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(F(-0.4) = \underline{0.3}\) | B1\(\checkmark\) (1) | \(\alpha < 1.0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(X) = (-2 \times 0.1) + \cdots + (3 \times 0.1)\) | M1 | Attempt at \(\sum x \cdot P(X=x)\); \(y = 3x+4\) |
| \(= \underline{0.3}\) | A1 | \(-2, 1, 47, 10, 13\) MIAM |
| \(\therefore E(3X+4) = (3 \times 0.3) + 4\) | M1 | \(F(Y) = 4.9\) MIAM; use of \(E(aX+b)\) |
| \(= \underline{4.9}\) | A1 (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Var}(X) = (-2^2 \times 0.1) + \cdots + (3^2 \times 0.1) - (0.3)^2\) | M1 | Attempt at \(\sum x^2 P(X=x) - \mu^2\); \(y = 2x+3\) |
| \(= \underline{2.01}\) | A1 | \(E(Y) = 3.6\); \(E(Y^2) = 21\); \(\text{Var}(Y) = 21 - (3.6)^2 = 8.04\) MIAM |
| \(\text{Var}(2X+3) = 4\text{Var}(X) = 4 \times 2.01\) | M1 | Use of \(\text{Var}(aX+b)\) |
| \(= \underline{8.04}\) | A1 (4) |
## Question 4:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $w = \underline{0.2}$ | B1 (1) | cao |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(-1 \leq X \leq 2) = P(0) + P(1) + P(2)$ | M1 | |
| $= \underline{0.6}$ | A1 (2) | |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $F(-0.4) = \underline{0.3}$ | B1$\checkmark$ (1) | $\alpha < 1.0$ |
### Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(X) = (-2 \times 0.1) + \cdots + (3 \times 0.1)$ | M1 | Attempt at $\sum x \cdot P(X=x)$; $y = 3x+4$ |
| $= \underline{0.3}$ | A1 | $-2, 1, 47, 10, 13$ MIAM |
| $\therefore E(3X+4) = (3 \times 0.3) + 4$ | M1 | $F(Y) = 4.9$ MIAM; use of $E(aX+b)$ |
| $= \underline{4.9}$ | A1 (4) | |
### Part (e)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Var}(X) = (-2^2 \times 0.1) + \cdots + (3^2 \times 0.1) - (0.3)^2$ | M1 | Attempt at $\sum x^2 P(X=x) - \mu^2$; $y = 2x+3$ |
| $= \underline{2.01}$ | A1 | $E(Y) = 3.6$; $E(Y^2) = 21$; $\text{Var}(Y) = 21 - (3.6)^2 = 8.04$ MIAM |
| $\text{Var}(2X+3) = 4\text{Var}(X) = 4 \times 2.01$ | M1 | Use of $\text{Var}(aX+b)$ |
| $= \underline{8.04}$ | A1 (4) | |
---4. The discrete random variable $X$ has the probability function shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$x$ & - 2 & - 1 & 0 & 1 & 2 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & 0.1 & $\alpha$ & 0.3 & 0.2 & 0.1 & 0.1 \\
\hline
\end{tabular}
\end{center}
Find
\begin{enumerate}[label=(\alph*)]
\item $\alpha$,
\item $\mathrm { P } ( - 1 < X \leq 2 )$,
\item $\mathrm { F } ( - 0.4 )$,
\item $\mathrm { E } ( 3 X + 4 )$,
\item $\operatorname { Var } ( 2 X + 3 )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2001 Q4 [12]}}