| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | October |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.3 This is a standard S1 probability distribution question requiring systematic application of well-known formulas (sum of probabilities = 1, E(X), E(X²), Var(X) = E(X²) - [E(X)]²) and linear transformations. The two-unknown system is straightforward to solve, and all parts follow predictable textbook patterns with no novel problem-solving required. Slightly easier than average due to its routine, methodical nature. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(x\) | - 2 | - 1 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(b\) | \(a\) | \(a\) | \(b\) | \(\frac { 1 } { 5 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| 2(a) | \(2a + 2b + 0.2 = 1\) (o.e.) | B1 |
| 2(b) | \(E(X) = -2b - a + a + 2b + 3 \times 0.2 = 0.6\) | M1, A1 |
| 2(c)(i) | \(E(X^2) = 3.5 \Rightarrow 3.5 = (-2)^2 \times b + (-1)^2 \times a + a + 2^2 \times b + 3^2 \times \frac{1}{5}\) or \(3.5 = 8b + 2a + 1.8\) or \(8b + 2a = 1.7\) (o.e.) | M1, A1 |
| 2(c)(ii) | Solving: \(2a + 2b = 0.8\) and \(2a + 8b = 1.7\) gives \(6b = 0.9\). So \(a = 0.25\) and \(b = 0.15\) | M1, A1A1 |
| 2(d) | \(\text{Var}(X) = 3.5 - "0.6"^2 = 3.5 - 0.36 = 3.14\) | M1, A1 |
| 2(e) | \(P(5 - 3X > 0) = P(5 > 3X) = P(X < 1.66\ldots)\) i.e. \(P(X = 1 \text{ or } -1 \text{ or } -2) = 0.65\) | M1, A1, A1ft |
| 2(f)(i) | \(E(Y) = 5 - 3E(X) = 5 - 1.8 = 3.2\) (Allow any exact equivalent e.g. \(\frac{16}{5}\)) | B1 |
| 2(f)(ii) | \(\text{Var}(Y) = (-3)^2 \text{Var}(X) = [9 \times 3.14] = 28.26\) or \(28.3\) or \(\frac{1413}{50}\) | M1, A1 |
**2(a)** | $2a + 2b + 0.2 = 1$ (o.e.) | B1 |
**2(b)** | $E(X) = -2b - a + a + 2b + 3 \times 0.2 = 0.6$ | M1, A1 | M1 for attempt at E(X) i.e. an expression with at least 4 correct terms; A1 for 0.6 or any exact equivalent. Allow 2/2 for 0.6 only or 0.6 following no incorrect working
**2(c)(i)** | $E(X^2) = 3.5 \Rightarrow 3.5 = (-2)^2 \times b + (-1)^2 \times a + a + 2^2 \times b + 3^2 \times \frac{1}{5}$ or $3.5 = 8b + 2a + 1.8$ or $8b + 2a = 1.7$ (o.e.) | M1, A1 | 1st M1 for attempt to form a second linear equation in a and b using $E(X^2) \geq 3$ correct terms; 2nd M1 for attempt to solve their linear equations...reducing to an equation in one variable
**2(c)(ii)** | Solving: $2a + 2b = 0.8$ and $2a + 8b = 1.7$ gives $6b = 0.9$. So $a = 0.25$ and $b = 0.15$ | M1, A1A1 | M1 for correct solution process; 1st A1 for $a = 0.25$; 2nd A1 for $b = 0.15$
**2(d)** | $\text{Var}(X) = 3.5 - "0.6"^2 = 3.5 - 0.36 = 3.14$ | M1, A1 | M1 for correct expression for Var(X) ft their 0.6; A1 for 3.14 or exact equivalent e.g. $\frac{157}{50}$
**2(e)** | $P(5 - 3X > 0) = P(5 > 3X) = P(X < 1.66\ldots)$ i.e. $P(X = 1 \text{ or } -1 \text{ or } -2) = 0.65$ | M1, A1, A1ft | M1 for attempt to solve inequality leading to $X < 1.66\ldots$ Allow $5 > 3X$ leading to $X < 0.6$; A1 for identifying 3 correct values of X required; A1ft for 0.65 or ft their $2a + b$ ($a \neq b$)
**2(f)(i)** | $E(Y) = 5 - 3E(X) = 5 - 1.8 = 3.2$ (Allow any exact equivalent e.g. $\frac{16}{5}$) | B1 |
**2(f)(ii)** | $\text{Var}(Y) = (-3)^2 \text{Var}(X) = [9 \times 3.14] = 28.26$ or $28.3$ or $\frac{1413}{50}$ | M1, A1 | M1 for seeing $(-3)^2 \text{Var}(X)$ or better [ft their value of Var(X)]; A1 for 28.26 or 28.3
---\begin{enumerate}
\item The discrete random variable $X$ has probability distribution
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & - 2 & - 1 & 1 & 2 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & $b$ & $a$ & $a$ & $b$ & $\frac { 1 } { 5 }$ \\
\hline
\end{tabular}
\end{center}
where $a$ and $b$ are constants.\\
(a) Write down an equation for $a$ and $b$.\\
(b) Calculate $\mathrm { E } ( X )$.
Given that $\mathrm { E } \left( X ^ { 2 } \right) = 3.5$\\
(c) (i) find a second equation in $a$ and $b$,\\
(ii) hence find the value of $a$ and the value of $b$.\\
(d) Find $\operatorname { Var } ( X )$.
The random variable $Y = 5 - 3 X$\\
(e) Find $\mathrm { P } ( Y > 0 )$.\\
(f) Find\\
(i) $\mathrm { E } ( Y )$,\\
(ii) $\operatorname { Var } ( Y )$.
\hfill \mbox{\textit{Edexcel S1 2016 Q2 [15]}}