| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Difficulty | Standard +0.3 This is a standard S2 question testing routine CDF/PDF manipulation and expectation calculations. Part (a) requires simple substitution, (b) is straightforward differentiation, (c) involves standard integration for E(X) and Var(X), and (d) requires finding the mode via differentiation. All techniques are textbook exercises with no novel problem-solving required, 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 integration |
The continuous random variable X has cumulative distribution function
$$\text{F}(x) = \begin{cases}
0, & x < 0, \\
\frac{1}{4}x²(4 - x²), & 0 \leq x \leq 1, \\
1, & x > 1.
\end{cases}$$
\begin{enumerate}[label=(\alph*)]
\item Find P(X > 0.7). [2]
\item Find the probability density function f(x) of X. [2]
\item Calculate E(X) and show that, to 3 decimal places, Var(X) = 0.057. [6]
\end{enumerate}
One measure of skewness is
$$\frac{\text{Mean} - \text{Mode}}{\text{Standard deviation}}$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Evaluate the skewness of the distribution of X. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [14]}}