5 The continuous random variable \(Y\) has probability density function given by
$$\mathrm { f } ( y ) = \begin{cases} \ln ( y ) & 1 \leqslant y \leqslant \mathrm { e }
0 & \text { otherwise } \end{cases}$$
- Verify that this is a valid probability density function.
- Show that the (cumulative) distribution function of \(Y\) is given by
$$\mathrm { F } ( y ) = \begin{cases} 0 & y < 1
y \ln y - y + 1 & 1 \leqslant y \leqslant \mathrm { e }
1 & \text { otherwise } \end{cases}$$ - Verify that the upper quartile of \(Y\) lies in the interval [2.45, 2.46].
- Find the (cumulative) distribution function of \(X\) where \(X = \ln Y\).