Probability Generating Functions

5 The discrete random variable \(U\) has probability distribution given by $$\mathrm { P } ( U = r ) = \begin{cases} \frac { 1 } { 16 } \binom { 4 } { r } & r = 0,1,2,3,4
0 & \text { otherwise } \end{cases}$$

1. Find and simplify the probability generating function (pgf) of \(U\).
2. Use the pgf to find \(\mathrm { E } ( U )\) and \(\operatorname { Var } ( U )\).
3. Identify the distribution of \(U\), giving the values of any parameters.
4. Obtain the pgf of \(Y\), where \(Y = U ^ { 2 }\).
5. State, giving a reason, whether you can obtain the pgf of \(U + Y\) by multiplying the pgf of \(U\) by the pgf of \(Y\).

4 The discrete random variable \(X\) has moment generating function \(\left( \frac { 1 } { 4 } + \frac { 3 } { 4 } \mathrm { e } ^ { t } \right) ^ { 3 }\).

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { P } ( X = 2 )\).
3. Show that \(X\) can be expressed as a sum of 3 independent observations of a random variable \(Y\). Obtain the probability distribution of \(Y\), and the variance of \(Y\).

1 The random variable \(X\) has the distribution \(\mathrm { B } ( n , p )\).

1. Show, from the definition, that the probability generating function of \(X\) is \(( q + p t ) ^ { n }\), where \(q = 1 - p\).
2. The independent random variable \(Y\) has the distribution \(\mathrm { B } ( 2 n , p )\) and \(T = X + Y\). Use probability generating functions to determine the distribution of \(T\), giving its parameters.

5 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 \(\frac { 1 } { 4 }\). The random variable \(X\) is the number of 6s obtained when dice \(A\) is thrown twice.

1. 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.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
3. Find \(\operatorname { Var } ( Z )\).
4. Use the probability generating function of \(Z\) to find 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 \(\mathrm { P } ( Y = 5 )\).

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. The discrete random variable \(X\) has probability generating function

$$\mathrm { G } _ { X } ( t ) = k \ln \left( \frac { 2 } { 2 - t } \right)$$ where \(k\) is a constant.

1. Find the exact value of \(k\)
2. Find the exact value of \(\operatorname { Var } ( X )\)
3. Find \(\mathrm { P } ( X = 3 )\)

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

1. Find \(\mathrm { E } ( \mathrm { Y } )\).
2. Write down an expression for the probability generating function of \(X\).
3. Find \(\mathrm { P } ( X = 4 )\).

3 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.

1. 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.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
3. Use the probability generating function of \(Z\) to find \(\operatorname { Var } ( Z )\).

4 The discrete random variable \(X\) has probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) given by $$G _ { X } ( t ) = 0.2 t + 0.5 t ^ { 2 } + 0.3 t ^ { 3 }$$ The random variable \(Y\) is the sum of two independent observations of \(X\).

1. Find the probability generating function of \(Y\), giving your answer as an expanded polynomial in \(t\). [3]
2. Use the probability generating function of \(Y\) to find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).

1. The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) where

$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { \sqrt { 4 - 3 t } }$$

1. Use calculus to find \(\operatorname { Var } ( X )\) Show your working clearly.
2. Find the exact value of \(\mathrm { P } ( X \leqslant 2 )\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\) The random variable \(Y = X _ { 1 } + X _ { 2 } + 1\)
3. By finding the probability generating function of \(Y\), state the name of the distribution of \(Y\)
4. Hence, or otherwise, find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)\)

7 The discrete random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t ) = \frac { 1 } { 126 } t \left( 64 - t ^ { 6 } \right) \left( 1 - \frac { t } { 2 } \right) ^ { - 1 }\).

1. Find \(\mathrm { P } ( Y = 3 )\).
2. Find \(\mathrm { E } ( Y )\).

1. The discrete random variable \(X\) has probability generating function

$$\mathrm { G } _ { X } ( t ) = \frac { t ^ { 2 } } { ( 3 - 2 t ) ^ { 2 } }$$

1. Specify the distribution of \(X\) A fair die is rolled repeatedly.
2. Describe an outcome that could be modelled by the random variable \(X\)
3. Use calculus and \(\mathrm { G } _ { X } ( t )\) to find
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) has probability generating function $$\mathrm { G } _ { Y } ( t ) = \frac { t ^ { 10 } } { \left( 3 - 2 t ^ { 3 } \right) ^ { 2 } }$$
4. Find the exact value of \(\mathrm { P } ( Y = 19 )\)