Solve for parameters using PGF coefficients

3 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 \(\mathrm { b } < \mathrm { a }\). The probability generating function of \(X\), the number of heads obtained by George, is \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\). The coefficients of \(t\) and \(t ^ { 2 }\) in \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) are \(\frac { 5 } { 12 }\) and \(\frac { 1 } { 12 }\) respectively.

1. Find the value of \(a\).
   The random variable \(Y\) is the sum of two independent observations of \(X\).
2. Find the probability generating function of \(Y\), giving your answer as a polynomial in \(t\).
3. Find \(\operatorname { Var } ( Y )\).

2 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 5 } + p t + q t ^ { 2 }$$ where \(p\) and \(q\) are constants.

1. Given that \(\mathrm { E } ( X ) = 1.1\), find the numerical value of \(\operatorname { Var } ( X )\).
   \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-04_2714_38_109_2010} The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t )\) given by $$\mathrm { G } _ { Y } ( t ) = \frac { 2 } { 3 } t \left( 1 + \frac { 1 } { 2 } t ^ { 2 } \right)$$ The random variable \(Z\) is the sum of independent observations of \(X\) and \(Y\).
2. Find the probability generating function of \(Z\).
3. Find \(\mathrm { P } ( Z > 2 )\).
4. State the most probable value of \(Z\).

4 The probability generating function of the discrete random variable \(Y\) is given by $$\mathrm { G } _ { Y } ( t ) = \frac { a + b t ^ { 3 } } { t }$$ where \(a\) and \(b\) are constants.

1. Given that \(\mathrm { E } ( Y ) = - 0.7\), find the values of \(a\) and \(b\).
2. Find \(\operatorname { Var } ( Y )\).
3. Find the probability that the sum of 10 random observations of \(Y\) is - 7 .

1. A discrete random variable \(X\) has probability generating function given by

$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 64 } \left( a + b t ^ { 2 } \right) ^ { 2 }$$ where \(a\) and \(b\) are positive constants.

1. Write down the value of \(\mathrm { P } ( X = 3 )\) Given that \(\mathrm { P } ( X = 4 ) = \frac { 25 } { 64 }\)
2. 1. find \(\mathrm { P } ( X = 2 )\)
   2. find \(\mathrm { E } ( X )\) The random variable \(Y = 3 X + 2\)
3. Find the probability generating function of \(Y\)