| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Discrete CDF to PMF |
| Difficulty | Easy -1.3 This is a straightforward S1 question testing basic understanding of CDFs. Part (a) requires simple subtraction of consecutive CDF values, part (b) is direct reading from the table, and part (c) tests understanding that F is constant between discrete values. All parts are routine recall/application with no problem-solving required. |
| Spec | 2.04a Discrete probability distributions |
| \(x\) | 1 | 3 | 5 | 7 |
| \(\mathrm {~F} ( x )\) | 0.2 | 0.5 | 0.9 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P(X = 1) = F(1) = 0.2\) e.g. \(P(X = 3) = F(3) - F(1) = 0.3\) | B1, M1 (4) | |
| x | 1 | 3 |
| \(P(X = x)\) | 0.2 | 0.3 |
| A1 A1 | 1st A1 for any two correct probabilities from \(P(X = 3) = 0.3, P(X = 5) = 0.4, P(X = 7) = 0.1\). 2nd A1 for fully correct probability distribution. For both A marks, condone missing/incorrect labels, but the probabilities must be associated with the correct \(x\)-values. | |
| (b) \(P(2 < X \le 6) = P(X = 3) + P(X = 5)\) \(= 0.7\) | M1, A1 (2) | For \(P(X = 3) + P(X = 5)\) (may ft their values) or \(F(5) - F(1)\). For 0.7 oe. |
| (c) \(F(4) = P(X \le 4) = P(X \le 3) = F(3) = 0.5\) | B1 (1) | For 0.5 oe. [7] |
**(a)** $P(X = 1) = F(1) = 0.2$ e.g. $P(X = 3) = F(3) - F(1) = 0.3$ | B1, M1 (4) |
| x | 1 | 3 | 5 | 7 |
| $P(X = x)$ | 0.2 | 0.3 | 0.4 | 0.1 |
| | | | | | | A1 A1 | | 1st A1 for any two correct probabilities from $P(X = 3) = 0.3, P(X = 5) = 0.4, P(X = 7) = 0.1$. 2nd A1 for fully correct probability distribution. For both A marks, condone missing/incorrect labels, but the probabilities must be associated with the correct $x$-values.
**(b)** $P(2 < X \le 6) = P(X = 3) + P(X = 5)$ $= 0.7$ | M1, A1 (2) | For $P(X = 3) + P(X = 5)$ (may ft their values) or $F(5) - F(1)$. For 0.7 oe.
**(c)** $F(4) = P(X \le 4) = P(X \le 3) = F(3) = 0.5$ | B1 (1) | For 0.5 oe. [7]
---\begin{enumerate}
\item A biased four-sided die has faces marked $1,3,5$ and 7 . The random variable $X$ represents the score on the die when it is rolled. The cumulative distribution function of $X , \mathrm {~F} ( x )$, is given in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 1 & 3 & 5 & 7 \\
\hline
$\mathrm {~F} ( x )$ & 0.2 & 0.5 & 0.9 & 1 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the probability distribution of $X$
\item Find $\mathrm { P } ( 2 < X \leqslant 6 )$
\item Write down the value of $\mathrm { F } ( 4 )$\\
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2014 Q3 [7]}}