| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | PDF from CDF |
| Difficulty | Standard +0.3 This is a standard S2 question testing routine CDF/PDF manipulation. Parts (a)-(d) involve straightforward differentiation, finding stationary points, sketching, and integration. Parts (e)-(f) require simple substitution and comparison. While multi-part with 14 marks total, each step follows textbook procedures without requiring novel insight or complex problem-solving, making it slightly easier than average. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
The continuous random variable X has cumulative distribution function F(x) given by
$$\text{F}(x) = \begin{cases}
0, & x < 1 \\
\frac{1}{2}(-x^3 + 6x^2 - 5), & 1 \leq x \leq 4 \\
1, & x > 4
\end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find the probability density function f(x). [3]
\item Find the mode of X. [2]
\item Sketch f(x) for all values of x. [3]
\item Find the mean $\mu$ of X. [3]
\item Show that F($\mu$) > 0.5. [1]
\item Show that the median of X lies between the mode and the mean. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q6 [14]}}