| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| 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 question testing routine probability distribution concepts: solving simultaneous equations from sum=1 and E(X) conditions, calculating E(X²), using variance formula Var(aX+b), interpreting cumulative distribution functions, and basic independence. All techniques are textbook exercises with no novel insight required, making it slightly easier than average but still requiring multiple steps and careful arithmetic. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( X = x )\) | \(a\) | \(a\) | \(a\) | \(b\) | \(b\) | 0.3 |
| \(y\) | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm {~F} ( y )\) | \(\frac { 1 } { 10 }\) | \(\frac { 2 } { 10 }\) | \(3 k\) | \(4 k\) | \(5 k\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(3a + 2b = 0.7\) | M1 | For attempt at linear equation in \(a\) and \(b\) based on sum of probs \(= 1\) |
| \(a + 2a + 3a + 4b + 5b + 1.8 = 4.2\) or \(6a + 9b = 2.4\) | M1 | For attempt at second linear equation based on \(E(X) = 4.2\); allow one slip |
| \(5b = 1\), attempt to solve | M1 | Must reduce to linear equation in one variable |
| \(b = \mathbf{0.2}\) | B1 cao | |
| \(a = \mathbf{0.1}\) | B1 cao |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X^2) = 1\times0.1 + 2^2\times0.1 + 3^2\times0.1 + 4^2\times0.2 + 5^2\times0.2 + 6^2\times0.3 (= 20.4)\) | B1 cso | For fully correct expression; no incorrect work seen; allow \(14\times0.1 + 41\times0.2 + 36\times0.3\) or \(0.1+0.4+0.9+3.2+5+10.8\) |
| Answer | Marks | Guidance |
|---|---|---|
| \([\text{Var}(X) =] \ 20.4 - 4.2^2 [= 2.76]\) | M1 | For correct expression for Var\((X)\); must see \(-4.2^2\) |
| \(\text{Var}(5-3X) = 9\text{Var}(X)\) | M1 | For \((-3)^2\text{Var}(X)\) or better; no need for a value; accept \(-3^2\) if clearly used as \(+9\) later |
| \(= \mathbf{24.84}\) or \(\mathbf{24.8}\) (allow \(\frac{621}{25}\)) | A1 cao |
| Answer | Marks |
|---|---|
| \([5k = 1 \text{ so}] \ k = \mathbf{0.2}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(Y=1) = 0.1\) | B1 | |
| e.g. \(P(Y=2) = F(2) - F(1) = 0.1\) | M1 | For correct use of \(F(y)\) to find one other prob; can ft their \(k\) if finding \(P(Y=y)\) for \(y > 2\) |
| Table: \(y\): 1, 2, 3, 4, 5; \(P(Y=y)\): 0.1, 0.1, 0.4, 0.2, 0.2 | A1 | For fully correct probability distribution; condone use of \(X(x)\) instead of \(Y(y)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=1) \times P(Y=1) = \mathbf{0.01}\) | M1, A1 cao | For correct expression or answer ft their \(P(Y=1)\) and \(P(X=1)\); for 0.01 or exact equivalent only; \(0.1\times0.1+0.1\times0.1\) or \(2\times0.1\times0.1\) are M0A0 |
# Question 5:
## Part (a)
| $3a + 2b = 0.7$ | M1 | For attempt at linear equation in $a$ and $b$ based on sum of probs $= 1$ |
| $a + 2a + 3a + 4b + 5b + 1.8 = 4.2$ or $6a + 9b = 2.4$ | M1 | For attempt at second linear equation based on $E(X) = 4.2$; allow one slip |
| $5b = 1$, attempt to solve | M1 | Must reduce to linear equation in one variable |
| $b = \mathbf{0.2}$ | B1 cao | |
| $a = \mathbf{0.1}$ | B1 cao | |
## Part (b)
| $E(X^2) = 1\times0.1 + 2^2\times0.1 + 3^2\times0.1 + 4^2\times0.2 + 5^2\times0.2 + 6^2\times0.3 (= 20.4)$ | B1 cso | For fully correct expression; no incorrect work seen; allow $14\times0.1 + 41\times0.2 + 36\times0.3$ or $0.1+0.4+0.9+3.2+5+10.8$ |
## Part (c)
| $[\text{Var}(X) =] \ 20.4 - 4.2^2 [= 2.76]$ | M1 | For correct expression for Var$(X)$; must see $-4.2^2$ |
| $\text{Var}(5-3X) = 9\text{Var}(X)$ | M1 | For $(-3)^2\text{Var}(X)$ or better; no need for a value; accept $-3^2$ if clearly used as $+9$ later |
| $= \mathbf{24.84}$ or $\mathbf{24.8}$ (allow $\frac{621}{25}$) | A1 cao | |
## Part (d)
| $[5k = 1 \text{ so}] \ k = \mathbf{0.2}$ | B1 | |
## Part (e)
| $P(Y=1) = 0.1$ | B1 | |
| e.g. $P(Y=2) = F(2) - F(1) = 0.1$ | M1 | For correct use of $F(y)$ to find one other prob; can ft their $k$ if finding $P(Y=y)$ for $y > 2$ |
| Table: $y$: 1, 2, 3, 4, 5; $P(Y=y)$: 0.1, 0.1, 0.4, 0.2, 0.2 | A1 | For fully correct probability distribution; condone use of $X(x)$ instead of $Y(y)$ |
## Part (f)
| $P(X=1) \times P(Y=1) = \mathbf{0.01}$ | M1, A1 cao | For correct expression or answer ft their $P(Y=1)$ and $P(X=1)$; for 0.01 or exact equivalent only; $0.1\times0.1+0.1\times0.1$ or $2\times0.1\times0.1$ are M0A0 |
---\begin{enumerate}
\item A biased die with six faces is rolled. The discrete random variable $X$ represents the score on the uppermost face. The probability distribution of $X$ is shown in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
$\mathrm { P } ( X = x )$ & $a$ & $a$ & $a$ & $b$ & $b$ & 0.3 \\
\hline
\end{tabular}
\end{center}
(a) Given that $\mathrm { E } ( X ) = 4.2$ find the value of $a$ and the value of $b$.\\
(b) Show that $\mathrm { E } \left( X ^ { 2 } \right) = 20.4$\\
(c) Find $\operatorname { Var } ( 5 - 3 X )$
A biased die with five faces is rolled. The discrete random variable $Y$ represents the score which is uppermost. The cumulative distribution function of $Y$ is shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$y$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$\mathrm {~F} ( y )$ & $\frac { 1 } { 10 }$ & $\frac { 2 } { 10 }$ & $3 k$ & $4 k$ & $5 k$ \\
\hline
\end{tabular}
\end{center}
(d) Find the value of $k$.\\
(e) Find the probability distribution of $Y$.
Each die is rolled once. The scores on the two dice are independent.\\
(f) Find the probability that the sum of the two scores equals 2\\
\hfill \mbox{\textit{Edexcel S1 2013 Q5 [15]}}