Given PGF manipulation and properties

Questions where a PGF formula is given and you must find constants, probabilities, variance, or transform to find PGF of related variables using algebraic manipulation.

5 questions · Standard +0.9

5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables
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CAIE Further Paper 4 2021 November Q5
9 marks Standard +0.3
5 The random variable \(X\) is such that \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 }\) for \(r = 1,2,3,4\), where \(k\) is a constant.
  1. Find the value of \(k\).
  2. Find the probability generating function \(\mathrm { G } _ { X } ( \mathrm { t } )\) of \(X\).
    The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } ) = \frac { 1 } { 4 } + \frac { 1 } { 2 } \mathrm { t } + \frac { 1 } { 4 } \mathrm { t } ^ { 2 }\).
    The random variable \(Z\) is the sum of \(X\) and \(Y\).
  3. Assuming that \(X\) and \(Y\) are independent, find the probability generating function \(\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )\) of \(Z\) as a polynomial in \(t\).
  4. Given that \(\mathrm { E } ( \mathrm { Z } ) = \frac { 13 } { 3 }\), use \(\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )\) to find \(\operatorname { Var } ( \mathrm { Z } )\).
Edexcel FS1 2019 June Q6
12 marks Challenging +1.2
  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 \(\mathrm { P } ( X = 3 )\)
Edexcel FS1 2021 June Q6
14 marks Standard +0.8
  1. The probability generating function of the random variable \(X\) is
$$\mathrm { G } _ { X } ( t ) = k ( 1 + 2 t ) ^ { 5 }$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 243 }\)
  2. Find \(\mathrm { P } ( X = 2 )\)
  3. Find the probability generating function of \(W = 2 X + 3\) The probability generating function of the random variable \(Y\) is $$\mathrm { G } _ { Y } ( t ) = \frac { t ( 1 + 2 t ) ^ { 2 } } { 9 }$$ Given that \(X\) and \(Y\) are independent,
  4. find the probability generating function of \(U = X + Y\) in its simplest form.
  5. Use calculus to find the value of \(\operatorname { Var } ( U )\)
Edexcel FS1 Specimen Q6
14 marks Standard +0.8
  1. The probability generating function of the discrete random variable \(X\) is given by
$$G _ { x } ( t ) = k \left( 3 + t + 2 t ^ { 2 } \right) ^ { 2 }$$
  1. Show that \(\mathrm { k } = \frac { 1 } { 36 }\)
  2. Find \(\mathrm { P } ( \mathrm { X } = 3 )\)
  3. Show that \(\operatorname { Var } ( \mathrm { X } ) = \frac { 29 } { 18 }\)
  4. Find the probability generating function of \(2 \mathrm { X } + 1\) \section*{Q uestion 6 continued} \section*{Q uestion 6 continued} \section*{Q uestion 6 continued}
WJEC Further Unit 5 Specimen Q7
17 marks Challenging +1.3
The discrete random variable \(X\) has the following probability distribution, where \(\theta\) is an unknown parameter belonging to the interval \(\left(0, \frac{1}{3}\right)\).
Value of \(X\)135
Probability\(\theta\)\(1 - 3\theta\)\(2\theta\)
  1. Obtain an expression for \(E(X)\) in terms of \(\theta\) and show that $$\text{Var}(X) = 4\theta(3 - \theta).$$ [4] In order to estimate the value of \(\theta\), a random sample of \(n\) observations on \(X\) was obtained and \(\bar{X}\) denotes the sample mean.
    1. Show that $$V = \frac{\bar{X} - 3}{2}$$ is an unbiased estimator for \(\theta\).
    2. Find an expression for the variance of \(V\). [4]
  3. Let \(Y\) denote the number of observations in the random sample that are equal to 1. Show that $$W = \frac{Y}{n}$$ is an unbiased estimator for \(\theta\) and find an expression for \(\text{Var}(W)\). [5]
  4. Determine which of \(V\) and \(W\) is the better estimator, explaining your method clearly. [4]