The continuous random variable \(T\) has probability density function given by
$$\mathrm { f } ( t ) = \begin{cases} 0 & t < 2
\frac { 2 } { ( t - 1 ) ^ { 3 } } & t \geqslant 2 \end{cases}$$
- Find the distribution function of \(T\), and find also \(\mathrm { P } ( T > 5 )\).
- Consecutive independent observations of \(T\) are made until the first observation that exceeds 5 is obtained. The random variable \(N\) is the total number of observations that have been made up to and including the observation exceeding 5. Find \(\mathrm { P } ( N > \mathrm { E } ( N ) )\).
- Find the probability density function of \(Y\), where \(Y = \frac { 1 } { T - 1 }\).