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 \() = \frac { 62 } { 125 }\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

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 \(a\).