| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | One unknown from sum constraint only |
| Difficulty | Easy -1.3 This is a straightforward S1 question testing basic probability distribution properties. Part (a) uses the sum-to-1 constraint (simple algebra), parts (b)-(d) are direct applications of standard formulas for expectation and variance, and part (e) requires solving a simple inequality then summing probabilities. All steps are routine recall with minimal problem-solving. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| \(x\) | - 1 | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 5 }\) | \(a\) | \(\frac { 1 } { 10 }\) | \(a\) | \(\frac { 1 } { 5 }\) |
**Question 3(a):** M1 for a clear attempt to use $\sum P(X = x) = 1$. Correct answer only 2/2. **NB Division by 5 in parts (b), (c) and (d) seen scores 0. Do not apply ISW.**
**Question 3(c):** 1st M1 for attempting $\sum x^2 P(X = x)$ at least two terms correct. Can follow through. 2nd M1 for attempting $E(X^2) - [E(X)]^2$ or allow subtracting 1 from their attempt at $E(X^2)$ provided no incorrect formula seen. Correct answer only 3/3.
**Question 3(d):** M1 for $(-2)^2 \text{Var}(X)$ or 4Var(X). Condone missing brackets provided final answer correct for their Var(X). Correct answer only 2/2.
**Question 3(e):** Allow M1 for distribution of $Y = 6 - 2X$ and correct attempt at $E(Y^2) - [E(Y)]^2$. M1 for identifying $X = 2, 3$. 1st A1ft for attempting to find their P(X=2) + P(X = 3). 2nd A1 for $\frac{9}{20}$ or 0.45.3. The discrete random variable $X$ has probability distribution given by
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 1 & 0 & 1 & 2 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & $\frac { 1 } { 5 }$ & $a$ & $\frac { 1 } { 10 }$ & $a$ & $\frac { 1 } { 5 }$ \\
\hline
\end{tabular}
\end{center}
where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.
\item Write down $\mathrm { E } ( X )$.
\item Find $\operatorname { Var } ( X )$.
The random variable $Y = 6 - 2 X$
\item Find $\operatorname { Var } ( Y )$.
\item Calculate $\mathrm { P } ( X \geqslant Y )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2010 Q3 [11]}}