| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate E(X) from cumulative distribution |
| Difficulty | Moderate -0.8 This is a straightforward S1 question testing basic understanding of cumulative distribution functions and expectation. Part (a) requires simple subtraction of consecutive F(Y) values, parts (b-d) are direct applications of standard formulas, and part (e) uses the routine Var(Y) = E(Y²) - [E(Y)]² formula. All steps are mechanical with no problem-solving or insight required. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf |
| \(y\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathrm {~F} ( Y )\) | 0.05 | 0.15 | 0.35 | 0.75 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(y\) | 0 |
| \(P(Y = y)\) | 0.05 | 0.1 |
| (b) \(0.1 + 0.2 = 0.3\) | M1 A1 | |
| (c) \(\sum yP(y) = 0 + 0.1 + 0.4 + 1.2 + 1 = 2.7\) | M1 A1 | |
| (d) \((2 \times 2.7) + 4 = 9.4\) | M1 A1 | |
| (e) \(E(Y^2) = \sum y^2P(y) = 0 + 0.1 + 0.8 + 3.6 + 4 = 8.5\) and \(\text{Var}(Y) = 8.5 - (2.7)^2 = 1.21\) | M1 A1 | Total 12 marks |
**(a)** | $y$ | 0 | 1 | 2 | 3 | 4 |
| $P(Y = y)$ | 0.05 | 0.1 | 0.2 | 0.4 | 0.25 | M1 A1 |
**(b)** $0.1 + 0.2 = 0.3$ | M1 A1 |
**(c)** $\sum yP(y) = 0 + 0.1 + 0.4 + 1.2 + 1 = 2.7$ | M1 A1 |
**(d)** $(2 \times 2.7) + 4 = 9.4$ | M1 A1 |
**(e)** $E(Y^2) = \sum y^2P(y) = 0 + 0.1 + 0.8 + 3.6 + 4 = 8.5$ and $\text{Var}(Y) = 8.5 - (2.7)^2 = 1.21$ | M1 A1 | Total 12 marks
---5. The discrete random variable $Y$ has the following cumulative distribution function.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$y$ & 0 & 1 & 2 & 3 & 4 \\
\hline
$\mathrm {~F} ( Y )$ & 0.05 & 0.15 & 0.35 & 0.75 & 1 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Write down the probability distribution of $Y$.
\item Find $\mathrm { P } ( 1 \leq Y < 3 )$.
\item Show that $\mathrm { E } ( Y ) = 2.7$
\item Find $\mathrm { E } ( 2 Y + 4 )$.
\item Find $\operatorname { Var } ( Y )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q5 [12]}}