Probability Generating Functions (PGFs) (Edexcel A Level Further Maths: Further Statistics 1)

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Author

Mark

Expertise

Maths

Constructing PGFs

What are Probability Generating Functions (PGFs)?

A probability generating function, $G_{X} (t)$ , is a polynomial in $t$
- The powers of $t$ are the values of $X$
- The coefficients are the corresponding probabilities of $X$
For example:
- $x$ 0 1 4 5
  
  $P (X = x)$ 0.4 0.3 0.2 0.1
- The PGF is $G_{X} (t) = 0.4 t^{0} + 0.3 t^{1} + 0.2 t^{4} + 0.1 t^{5}$
  This simplifies to $G_{X} (t) = \frac{1}{10} (4 + 3 t + 2 t^{4} + t^{5})$
The variable $t$ is called a dummy variable
- It is used to create a polynomial structure
- Do not confuse it with $X$
Coefficients can never be negative
- They are probabilities!

What is the value of G(1)?

$G_{X} (1) = 1$ always
This is because substituting $t = 1$ into a PGF:
- Turns all powers of $t$ into 1
- Leaves the sum of all probabilities which equals 1
- For example
  $G_{X} (1) = 0.4 + 0.3 + 0.2 + 0.1 = 1$

What is E(t^X)?

$G_{X} (t) = E (t^{X})$ is the formal definition of a PGF given in the Formulae Booklet
- Recall that $E (X) = \sum_{} x P (X = x)$ is the expectation of $X$
  - Th expectation of a function of $X$ is $E (g (X)) = \sum_{} g (x) P (X = x)$
- Choosing the function to be gives $E (t^{X}) = \sum_{} t^{x} P (X = x)$
  - This is the sum of powers of $t$ multiplied by their corresponding probabilities
  - That is the probability generating function of $X$

Exam Tip

Don't forget to use $G_{X} (1) = 1$ in harder algebraic questions!

Worked example

A discrete random variable, $X$ , is given by the probability distribution below.

$x$	3	4	6	10	11
$P (X = x)$	$\frac{1}{8}$	$\frac{1}{16}$	$\frac{1}{4}$	$p$	$\frac{3}{8}$

Find the probability generating function of $X$ .

constructing-pgfs

Finding Probabilities from PGFs

How do I find probabilities from PGFs?

Fully expand the PGF
- $\frac{1}{4} {(1 + t)}^{2}$ expands to $\frac{1}{4} + \frac{1}{2} t + \frac{1}{4} t^{2}$
Read off the relevant coefficient
- $P (X = 2)$ is the coefficient of $t^{2}$
  - $P (X = 2) = \frac{1}{4}$
Remember $G (1) = 1$

When can I use the General Binomial Theorem?

When PGFs can be written in the form ${(1 + . . .)}^{n}$
- Where $n$ is a positive or negative rational number
- You may have to rearrange to get this form
Use the General Binomial Theorem to expand the PGF
- Simplify each term
- Read off probabilities

When can I use Maclaurin Series?

When a PGF is written as a function that is not a polynomial
For example
- $G_{X} (t) = - \frac{1}{\ln 2} \ln (1 - 0.5 t)$
- A Maclaurin Series is given by $\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - . . .$
- Use it to expand the PGF
  - $G_{X} (t) = - \frac{1}{\ln 2} ((- 0.5 t) - \frac{{(- 0.5 t)}^{2}}{2} + \frac{{(- 0.5 t)}^{3}}{3} - . . .)$
- Then simplify each term

Worked example

A probability generating function for a discrete random variable $X$ is given in the form

$G_{X} (t) = \frac{t^{2}}{{(5 - 4 t)}^{2}}$

Find $P (X = 4)$ , showing your working clearly.

finding-probabilities-from-pgfs

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$x$	0	1	4	5
$P (X = x)$	0.4	0.3	0.2	0.1