Cumulative distribution function
Cumulative distribution function
IB Year 5 | Grade 11
The continuous random variable \(T\) has probability density function given by
\[\mathrm{f}(t) = \begin{cases} k(t - 2), & \text{for}\hspace{0.5em} 2 \le t \le 4 \\ 0, & \text{otherwise} \end{cases}\]
Where \(k\) is a constant.
- Find, in terms of \(k\), the cumulative distribution function of \(T\).
- Hence or otherwise, find the value of \(k\).
- Show that \(\mathrm{P}(T > 3) = 0.75\).
- The event \(T > 3\) is denoted by \(A\), and the event \(2T > 3\) is denoted by \(B\). Find \(\mathrm{P}(A \cup B)\) and \(\mathrm{P}(A \cap B)\).
Solution:
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Answer:(a) \(\mathrm{F}(t)=\begin{cases}0,&t<2,\\ \frac{k(t-2)^2}{2},&2\leq t\leq4,\\ 2k,&t>4,\end{cases}\) where normalization requires \(2k=1\). (b) \(k=\frac12\). (c) \(\mathrm{P}(T>3)=0.75\). (d) \(\mathrm{P}(A\cup B)=1\), \(\mathrm{P}(A\cap B)=\frac34\).
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