Standard +0.3 This is a straightforward continuous probability distribution question requiring basic integration of a polynomial (part a) and geometric reasoning about a semicircular pdf (part b). Part (a) involves substituting limits into a simple integral. Part (b) uses symmetry properties and the normalization condition for pdfs. All techniques are standard S2 material with no novel problem-solving required, making it slightly easier than average.
The time for which Lucy has to wait at a certain traffic light each day is \(T\) minutes, where \(T\) has probability density function given by
$$f ( t ) = \begin{cases} \frac { 3 } { 2 } t - \frac { 3 } { 4 } t ^ { 2 } & 0 \leqslant t \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
Find the probability that, on a randomly chosen day, Lucy has to wait for less than half a minute at the traffic light.
\includegraphics[max width=\textwidth, alt={}, center]{c08d3228-430e-4158-9362-1655deb1feb7-2_405_791_1471_715}
The diagram shows the graph of the probability density function, g , of a random variable \(X\). The graph of g is a semicircle with centre \(( 0,0 )\) and radius \(a\). Elsewhere \(\mathrm { g } ( x ) = 0\).
Find the value of \(a\).
State the value of \(\mathrm { E } ( X )\).
Given that \(\mathrm { P } ( X < - c ) = 0.2\), find \(\mathrm { P } ( X < c )\).
3
\begin{enumerate}[label=(\alph*)]
\item The time for which Lucy has to wait at a certain traffic light each day is $T$ minutes, where $T$ has probability density function given by
$$f ( t ) = \begin{cases} \frac { 3 } { 2 } t - \frac { 3 } { 4 } t ^ { 2 } & 0 \leqslant t \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
Find the probability that, on a randomly chosen day, Lucy has to wait for less than half a minute at the traffic light.
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{c08d3228-430e-4158-9362-1655deb1feb7-2_405_791_1471_715}
The diagram shows the graph of the probability density function, g , of a random variable $X$. The graph of g is a semicircle with centre $( 0,0 )$ and radius $a$. Elsewhere $\mathrm { g } ( x ) = 0$.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $a$.
\item State the value of $\mathrm { E } ( X )$.
\item Given that $\mathrm { P } ( X < - c ) = 0.2$, find $\mathrm { P } ( X < c )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2014 Q3 [8]}}