- The probability generating function of the random variable \(X\) is
$$\mathrm { G } _ { X } ( t ) = k ( 1 + 2 t ) ^ { 5 }$$
where \(k\) is a constant.
- Show that \(k = \frac { 1 } { 243 }\)
- Find \(\mathrm { P } ( X = 2 )\)
- Find the probability generating function of \(W = 2 X + 3\)
The probability generating function of the random variable \(Y\) is
$$\mathrm { G } _ { Y } ( t ) = \frac { t ( 1 + 2 t ) ^ { 2 } } { 9 }$$
Given that \(X\) and \(Y\) are independent,
- find the probability generating function of \(U = X + Y\) in its simplest form.
- Use calculus to find the value of \(\operatorname { Var } ( U )\)