| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Continuous Probability Distributions and Random Variables |
| Type | Calculate probability P(X in interval) |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard CDF/PDF manipulation. Part (a) requires simple substitution into the CDF formula, part (b) is routine differentiation, and part (c) applies standard expectation/variance formulas to a uniform distribution. All steps are mechanical with no problem-solving insight 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 integration5.03e Find cdf: by integration5.03f Relate pdf-cdf: medians and percentiles |
4 The continuous random variable $X$ has the cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c }
0 & x < - c \\
\frac { x + c } { 4 c } & - c \leqslant x \leqslant 3 c \\
1 & x > 3 c
\end{array} \right.$$
where $c$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Determine $\mathrm { P } \left( - \frac { 3 c } { 4 } < X < \frac { 3 c } { 4 } \right)$.
\item Show that the probability density function, $\mathrm { f } ( x )$, of $X$ is
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { 1 } { 4 c } & - c \leqslant x \leqslant 3 c \\
0 & \text { otherwise }
\end{array} \right.$$
\item Hence, or otherwise, find expressions, in terms of $c$, for:
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { E } ( X )$;
\item $\operatorname { Var } ( X )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2009 Q4 [6]}}