5. The continuous random variable \(T\) has cumulative distribution function
$$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 ,
1 - \mathrm { e } ^ { - 0.25 t } & t \geqslant 0 . \end{cases}$$
- Find the cumulative distribution function of \(2 T\).
- Show that, for constant \(k , \quad \mathrm { E } \left( \mathrm { e } ^ { k T } \right) = \frac { 1 } { 1 - 4 k }\).
You should state with a reason the range of values of \(k\) for which this result is valid.
- \(T\) is the time before a certain event occurs.
Show that the probability that no event occurs between time \(T = 0\) and time \(T = \theta\) is the same as the probability that the value of a random variable with the distribution \(\operatorname { Po } ( \lambda )\) is 0 , for a certain value of \(\lambda\). You should state this value of \(\lambda\) in terms of \(\theta\).
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