The discrete random variable \(X\) has probability generating function \(G_X(t)\) given by

\(G_X(t)=\frac{t}{(3-2t)^2}\).

(a) Find \(\mathrm{E}(X)\) and \(\operatorname{Var}(X)\).

The discrete random variable \(Y\) has probability generating function \(G_Y(t)\) given by

\(G_Y(t)=\frac{t^2}{(3-2t)^2}\).

The random variable \(Z\) is the sum of the random variables \(X\) and \(Y\).

(b) Assuming \(X\) and \(Y\) are independent, find \(\mathrm{P}(Z\gt 4)\).

The random variable \(Y\) is the sum of two independent observations of the random variable \(X\). The probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\) is given by
\(\mathrm{G}_{Y}(t)=\frac{t^{2}}{(4-3 t)^{4}}\)
(a) Find \(E(Y)\).

(b) Write down an expression for the probability generating function of \(X\).

(c) Find \(\mathrm{P}(X=4)\).

The random variable \(X\) has probability generating function \(\mathrm{G}_{X}(t)\) given by
\(\mathrm{G}_{X}(t)=\operatorname{ct}(1+t)^{5}\)
where \(c\) is a constant.
(a) Find the value of \(c\).

(b) Find the value of \(\mathrm{E}(X)\).

The random variable \(Y\) is the sum of two independent values of \(X\).
(c) Write down the probability generating function of \(Y\) and hence find \(\operatorname{Var}(Y)\).

(d) Find \(\mathrm{P}(Y=5)\).

The random variable \(X\) has the geometric distribution \(\operatorname{Geo}(p)\).
(a) Show that the probability generating function of \(X\) is \(\frac{p t}{1-q t}\), where \(q=1-p\).
(b) Use the probability generating function of \(X\) to show that \(\operatorname{Var}(X)=\frac{q}{p^{2}}\).

Kenny throws an ordinary fair 6-sided dice repeatedly. The random variable \(X\) is the number of throws that Kenny takes in order to obtain a 6 . The random variable \(Z\) denotes the sum of two independent values of \(X\).
(c) Find the probability generating function of \(Z\).

Toby has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 3 marbles from the bag, without replacement. The random variable \(X\) is the number of red marbles that Toby obtains.
(a) Find the probability generating function of \(X\).

Ling also has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 2 marbles from his bag, without replacement. The random variable \(Y\) is the number of red marbles that Ling obtains. It is given that the probability generating function of \(Y\) is \(\frac{1}{12}\left(1+6 t+5 t^{2}\right)\).

The random variable \(Z\) is the total number of red marbles that Toby and Ling obtain.
(b) Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
(c) Use the probability generating function of \(Z\) to find \(\operatorname{Var}(Z)\).

The probability generating function, \(\mathrm{G}_{Y}(t)\), of the random variable \(Y\) is given by
\(\mathrm{G}_{Y}(t)=0.04+0.2 t+0.37 t^{2}+0.3 t^{3}+0.09 t^{4}\)
(a) Find \(\operatorname{Var}(Y)\).

The random variable \(Y\) is the sum of two independent observations of the random variable \(X\).
(b) Find the probability generating function of \(X\), giving your answer as a polynomial in \(t\).

George throws two coins, \(A\) and \(B\), at the same time. Coin \(A\) is biased so that the probability of obtaining a head is \(a\). Coin \(B\) is biased so that the probability of obtaining a head is \(b\), where \(b\lt a\). The probability generating function of \(X\), the number of heads obtained by George, is \(\mathrm{G}_{X}(t)\). The coefficients of \(t\) and \(t^{2}\) in \(\mathrm{G}_{X}(t)\) are \(\frac{5}{12}\) and \(\frac{1}{12}\) respectively.
(a) Find the value of \(a\).

The random variable \(Y\) is the sum of two independent observations of \(X\).
(b) Find the probability generating function of \(Y\), giving your answer as a polynomial in \(t\).

(c) Find \(\operatorname{Var}(Y)\).

A 6-sided dice, \(A\), with faces numbered 1, 2, 3, 4, 5, 6 is biased so that the probability of throwing a 6 is \(\frac14\). The random variable \(X\) is the number of 6s obtained when dice \(A\) is thrown twice.

(a) Find the probability generating function of \(X\).

A second dice, \(B\), with faces numbered 1, 2, 3, 4, 5, 6 is unbiased. The random variable \(Y\) is the number of 6s obtained when dice \(B\) is thrown twice.

The random variable \(Z\) is the total number of 6s obtained when both dice are thrown twice.

(b) Find the probability generating function of \(Z\), expressing your answer as a polynomial.

(c) Find \(\operatorname{Var}(Z)\).

(d) Use the probability generating function of \(Z\) to find the most probable value of \(Z\).

The discrete random variable \(X\) has probability generating function \(\mathrm{G}_{X}(t)\) given by
\(\mathrm{G}_{X}(t)=0.2 t+0.5 t^{2}+0.3 t^{3} .\)

The random variable \(Y\) is the sum of two independent observations of \(X\).
(a) Find the probability generating function of \(Y\), giving your answer as an expanded polynomial in \(t\).
(b) Use the probability generating function of \(Y\) to find \(\mathrm{E}(Y)\) and \(\operatorname{Var}(Y)\).