A discrete random variable \(X\) takes values \(r=0,1,2\) with probabilities \(\mathrm{P}(X=r)\) as given in the following table.

(a) Write down the probability generating function of \(X\), and use it to find an expression for \(\mathrm{E}(X)\) in terms of \(a\) and \(b\).

(b) Show that \(\operatorname{Var}(X)=2b+2(a+b)(1-2a-2b)\).

The random variable \(Y\) is defined by \(Y=X_1+X_2+X_3+\cdots+X_{10}\), where \(X_1,X_2,X_3,\ldots,X_{10}\) are ten independent observations of \(X\).

(c) Using the probability generating function of \(Y\), and your answer to part (a), show that \(\mathrm{E}(Y)=10\mathrm{E}(X)\).

(d) For the case \(b=0\), define fully the distribution of \(Y\).

The random variable \(X\) takes values 1 and 2 with probabilities \(\frac{2}{5}\) and \(\frac{3}{5}\) respectively. (a) Write down the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\). The random variable \(Y\) is the sum of four independent observations of \(X\). (b) Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\). Give your answer in the form \(\mathrm{G}_{Y}(t)=a t^{m}(b+c t)^{n}\), where \(a, b, c, m\) and \(n\) are constants to be determined. (c) Use \(\mathrm{G}_{Y}(t)\) to find \(\mathrm{P}(Y=6)\). (d) Find \(\operatorname{Var}(Y)\).