E(X) & Var(X) of PGFs (Edexcel A Level Further Maths): Revision Note

Author

Mark Curtis

Last updated

4 January 2025

E(X) of PGFs

How do I find E(X) of a PGF?

$E (X) = G_{X}' (1)$
- Differentiate the PGF and substitute in $t = 1$
This is because
- $G_{X} (t) = E (t^{X}) = \sum_{}^{} t^{x} P (X = x)$
- So $G_{X}' (t) = {\frac{d}{d t} [\sum_{}^{} t^{x} P (X = x)] = \sum}_{}^{} x t^{x - 1} P (X = x)$
- Substituting $t = 1$ gives $G_{X}' (1) = \sum_{}^{} x P (X = x) = E (X)$
You may need the chain, product or quotient rule.

Examiner Tips and Tricks

$E (X) = G'_{X} (1)$ is given in the Formulae Booklet

Worked Example

The probability generating function for a discrete random variable $X$ is given by

$G_{X} (t) = \frac{1}{81} t^{3} {(1 + 2 t)}^{4}$

Find $E (X)$ .

Var(X) of PGFs

How do I find Var(X) of a PGF?

The formula is $Var (X) = G_{X}'' (1) + G_{X}' (1) - {[G_{X}' (1)]}^{2}$
You may need the chain, product or quotient rule.
The first two terms in the formula are $E (X^{2})$
- $E (X^{2}) = G_{X}'' (1) + G_{X}' (1)$
The formula comes from
- $G_{X} (t) = E (t^{X}) = \sum_{}^{} t^{x} P (X = x)$
- So $G_{X}' (t) = {\frac{d}{d t} [\sum_{}^{} t^{x} P (X = x)] = \sum}_{}^{} x t^{x - 1} P (X = x)$
  - Recall $G_{X}' (1) = \sum x P (X = x) = E (X)$
- And $G_{X}'' (t) = {\frac{d}{d t} [\sum_{}^{} x t^{x - 1} P (X = x)] = \sum}_{}^{} x (x - 1) t^{x - 2} P (X = x)$
- Substituting $t = 1$ gives $G_{X}'' (1) = \sum_{}^{} x (x - 1) P (X = x) = E (X (X - 1)) = E (X^{2}) - E (X)$
- Make $E (X^{2})$ the subject
  - $E (X^{2}) = G_{X}'' (1) + E (X)$
- Then substitute it into $Var (X) = E (X^{2}) - {(E (X))}^{2}$
- And replace $E (X)$ with $G_{X}' (1)$

Examiner Tips and Tricks

$Var (X) = G_{X}'' (1) + G_{X}' (1) - {[G_{X}' (1)]}^{2}$ is given in the Formulae Booklet

Worked Example

The probability generating function for a discrete random variable, $X$ , is given by

$G_{X} (t) = 0.2 + 0.4 t + 0.3 t^{4} + 0.1 t^{5}$

Find $Var (X)$ .

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Previous:Probability Generating Functions (PGFs)Next:PGFs of Standard Distributions

E(X) & Var(X) of PGFs (Edexcel A Level Further Maths): Revision Note

E(X) of PGFs

How do I find E(X) of a PGF?

Var(X) of PGFs

How do I find Var(X) of a PGF?

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Discrete Probability Distributions

E(X) & Var(X) of Discrete Random Variables

Linear Transformations of DRVs

Problem Solving with DRVs

Poisson & Binomial Distributions

The Poisson Distribution

Poisson Approximations of Binomials

Geometric & Negative Binomial Distributions

The Geometric Distribution

The Negative Binomial Distribution

Probability Generating Functions

Probability Generating Functions

Probability Generating Functions (PGFs)

E(X) & Var(X) of PGFs

PGFs of Standard Distributions

PGFs of Sums & Transformations

Central Limit Theorem

Central Limit Theorem

Central Limit Theorem

Hypothesis Testing

Poisson & Geometric Hypothesis Testing

Poisson Hypothesis Testing

Geometric Hypothesis Testing

Chi Squared Tests

Goodness of Fit

Chi Squared Tests for Standard Distributions

Chi Squared Tests for Contingency Tables

Quality of Tests

Type I & Type II Errors

Size & Power of Test

Author: Mark Curtis