Determine constant in PGF

5 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( \mathrm { t } )\) given by $$\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } ) = \mathrm { k } \left( 1 + 3 \mathrm { t } + 4 \mathrm { t } ^ { 2 } \right)$$ where \(k\) is a constant.

1. Show that \(\mathrm { E } ( X ) = \frac { 11 } { 8 }\).
   The random variable \(Y\) has probability generating function \(\mathrm { G } _ { \gamma } ( \mathrm { t } )\) given by $$G _ { \gamma } ( t ) = \frac { 1 } { 3 } t ^ { 2 } ( 1 + 2 t )$$ The random variables \(X\) and \(Y\) are independent and \(\mathrm { Z } = \mathrm { X } + \mathrm { Y }\).
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
3. Use your answer to part (b) to find the value of \(\operatorname { Var } ( Z )\).
4. Write down the most probable value of \(Z\).

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.

1. Find the value of \(c\).
2. Find the value of \(\mathrm { E } ( X )\). \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-06_2718_33_141_2014} The random variable \(Y\) is the sum of two independent values of \(X\).
3. Write down the probability generating function of \(Y\) and hence find \(\operatorname { Var } ( Y )\).
4. Find \(\mathrm { P } ( Y = 5 )\).

7 The probability generating function of the random variable \(X\) is given by $$\mathrm { G } ( t ) = \frac { 1 + a t } { 4 - t }$$ where \(a\) is a constant.

1. Find the value of \(a\).
2. Find \(\mathrm { P } ( X = 3 )\). The sum of 3 independent observations of \(X\) is denoted by \(Y\). The probability generating function of \(Y\) is denoted by \(\mathrm { H } ( t )\).
3. Use \(\mathrm { H } ( t )\) to find \(\mathrm { E } ( Y )\).
4. By considering \(\mathrm { H } ( - 1 ) + \mathrm { H } ( 1 )\), show that \(\mathrm { P } ( Y\) is an even number \() = \frac { 62 } { 125 }\).

3 The discrete random variable \(X\) has probability generating function \(\frac { t } { a - b t }\), where \(a\) and \(b\) are constants.

1. Find a relationship between \(a\) and \(b\).
2. Use the probability generating function to find \(\mathrm { E } ( X )\) in terms of \(a\), giving your answer as simply as possible.
3. Expand the probability generating function as a power series, as far as the term in \(t ^ { 3 }\), giving the coefficients in terms of \(a\) and \(b\).
4. Name the distribution for which \(\frac { t } { a - b t }\) is the probability generating function, and state its parameter(s) in terms of \(a\).

5 The discrete random variable \(X\) has probability generating function given by $$\mathrm { G } _ { X } ( t ) = k \left( 5 t ^ { - 1 } + 3 + 2 t ^ { 2 } \right) ,$$ where \(k\) is a constant.

1. Find
   1. the value of \(k\),
   2. the modal value of \(X\).
   3. The random variables \(X _ { 1 }\) and \(X _ { 2 }\) are independent observations of \(X\).
      (a) Write down the probability generating function of \(Y\), where \(Y = X _ { 1 } + X _ { 2 }\).
      (b) Use your answer to part (ii)(a) to find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).

1 The discrete random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 } ,$$ where \(a\) is a constant.

1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
2. Find the value of \(\mathrm { E } ( X )\).

1 The discrete random variable X has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 } ,$$ where \(a\) is a constant.

1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
2. Find the value of \(\mathrm { E } ( X )\).

1 The discrete random variable X has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 }$$ where \(a\) is a constant.

1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
2. Find the value of \(\mathrm { E } ( X )\).