A LevelFurther Maths: Further Statistics 1EdexcelRevision NotesProbability Generating FunctionsProbability Generating FunctionsE(X) & Var(X) of PGFs

E(X) & Var(X) of PGFs (Edexcel A Level Further Maths: Further Statistics 1)

Revision Note

Author

Mark

Expertise

Maths

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.

Exam Tip

$E (X) = G'_{X} (1)$ 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 $E (X)$ .

ex-of-pgfs

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 $G_{X}' (1)$

Exam Tip

$Var (X) = G_{X}'' (1) + G_{X}' (1) - {[G_{X}' (1)]}^{2}$ 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 $Var (X)$ .

varx-of-pgfs

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Author: Mark

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.