| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Calculate and compare mean, median, mode |
| Difficulty | Standard +0.3 This is a standard S2 cumulative distribution function question testing routine techniques: finding k from continuity, calculating median/mode/mean, differentiating to find pdf, and sketching. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration |
A continuous random variable X has cumulative distribution function F(x) given by
$$\text{F}(x) = \begin{cases}
0, & x < 0, \\
kx^2 + 2kx, & 0 \leq x \leq 2, \\
8k, & x > 2.
\end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Show that $k = \frac{1}{8}$. [1]
\item Find the median of X. [3]
\item Find the probability density function f(x). [3]
\item Sketch f(x) for all values of x. [3]
\item Write down the mode of X. [1]
\item Find E(X). [3]
\item Comment on the skewness of this distribution. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q7 [16]}}