The random variable \(X\) has a discrete uniform distribution and takes the values \(1,2,3,4\) Find
\(\mathrm { F } ( 3 )\), where \(\mathrm { F } ( x )\) is the cumulative distribution function of \(X\),
\(\mathrm { E } ( X )\).
Show that \(\operatorname { Var } ( X ) = \frac { 5 } { 4 }\)
The random variable \(Y\) has a discrete uniform distribution and takes the values
$$3,3 + k , 3 + 2 k , 3 + 3 k$$
where \(k\) is a constant.
Write down \(\mathrm { P } ( Y = y )\) for \(y = 3,3 + k , 3 + 2 k , 3 + 3 k\)
The relationship between \(X\) and \(Y\) may be written in the form \(Y = k X + c\) where \(c\) is a constant.
Find \(\operatorname { Var } ( Y )\) in terms of \(k\).