| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | Conditional probability with CDF |
| Difficulty | Standard +0.3 This is a standard S2 question on continuous distributions requiring routine application of CDF properties: continuity conditions give two equations for a and k, median uses F(m)=0.5, and conditional probability uses P(A|B)=P(A∩B)/P(B). All steps are algorithmic with no novel insight required, making it slightly easier than average. |
| Spec | 2.03d Calculate conditional probability: from first principles5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(k(2-a)^2 = 0\), so \(a = 2\) | M1, A1 | |
| \(k(6-a)^2 = 1\), so \(k = \frac{1}{16}\) | M1, A1 | |
| (b) Need \(\frac{1}{16}(x-2)^2 = \frac{1}{4}\) | M1, A1 | |
| \(x - 2 = \pm\sqrt{8} = 2\sqrt{2}\) | M1, A1 | |
| \(x = 2 \pm 2\sqrt{2}\) | M1, A1 | |
| (c) \(P(X > 4) = 1 - F(4) = 1 - \frac{1}{4} = \frac{3}{4}\) | M1, A1 | |
| \(P(X > 5) = 1 - F(5) = 1 - \frac{9}{16} = \frac{7}{16}\) | M1, A1 | |
| \(P(X > 5 | X > 4) = \frac{7}{16} \div \frac{3}{4} = \frac{7}{12}\) | M1, A1 |
(a) $k(2-a)^2 = 0$, so $a = 2$ | M1, A1 |
$k(6-a)^2 = 1$, so $k = \frac{1}{16}$ | M1, A1 |
(b) Need $\frac{1}{16}(x-2)^2 = \frac{1}{4}$ | M1, A1 |
$x - 2 = \pm\sqrt{8} = 2\sqrt{2}$ | M1, A1 |
$x = 2 \pm 2\sqrt{2}$ | M1, A1 |
(c) $P(X > 4) = 1 - F(4) = 1 - \frac{1}{4} = \frac{3}{4}$ | M1, A1 |
$P(X > 5) = 1 - F(5) = 1 - \frac{9}{16} = \frac{7}{16}$ | M1, A1 |
$P(X > 5 | X > 4) = \frac{7}{16} \div \frac{3}{4} = \frac{7}{12}$ | M1, A1 | 14 marks total |A continuous random variable $X$ has the cumulative distribution function
$$F(x) = 0 \quad x < 2,$$
$$F(x) = k(x - a)^2 \quad 2 \leq x \leq 6,$$
$$F(x) = 1 \quad x \geq 6.$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $a$ and $k$. [4 marks]
\item Show that the median of $X$ is $2(1 + \sqrt{2})$. [4 marks]
\item Given that $X > 4$, find the probability that $X > 5$. [6 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q5 [14]}}