4. A discrete random variable \(X\) takes only positive integer values. It has a cumulative distribution function \(\mathrm { F } ( x ) = \mathrm { P } ( X \leq x )\) defined in the table below.
| \(X\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| \(\mathrm {~F} ( x )\) | 0.1 | 0.2 | 0.25 | 0.4 | 0.5 | 0.6 | 0.75 | 1 |
- Determine the probability function, \(\mathrm { P } ( X = x )\), of \(X\).
- Calculate \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = 5.76\).
- Given that \(Y = 2 X + 3\), find the mean and variance of \(Y\).