Probability Generating Functions Revision Notes for Edexcel A-Level Further Mathematics

20.1.3 PGFs of Sums & Transformations

Introduction

Probability Generating Functions (PGFs) are not only useful for analysing individual random variables but also for understanding the behaviour of sums and transformations of random variables. For example, the PGF of the sum of independent random variables is the product of their PGFs.

This note covers:

PGFs of sums of independent random variables.
Transformations of random variables using PGFs.
Worked examples to apply these principles.

PGFs of Sums of Independent Random Variables

If $X$ and $Y$ are independent random variables, and $Z = X + Y$ , the PGF of $Z$ is:

G_Z(t) = G_X(t) \times G_Y(t)

where:

$G_X(t)$ : PGF of $X$
$G_Y(t)$ : PGF of $Y$
$G_Z(t)$ : PGF of $Z$ This principle generalises to the sum of $n$ independent random variables:

G_{S_n}(t) = \prod_{i=1}^n G_{X_i}(t)

where $S_n = X_1 + X_2 + \dots + X_n$ and $X_i$ are independent.

Transformations of Random Variables

For transformations of a random variable $X$ :

Scaling: If $Z = aX$ , then:

G_Z(t) = G_X(t^a)

Shifted Sum: If $Z = X + c$ , where $c$ is a constant, then:

G_Z(t) = t^c G_X(t)

These transformations allow for adjustments to the PGF to account for scaling or shifts.

Mean and Variance of Sums

The properties of PGFs also apply to sums:

Mean of $Z = X + Y$

\mathbb{E}[Z] = \mathbb{E}[X] + \mathbb{E}[Y]

Variance of $Z = X + Y$ (independent $X$ and $Y$ )

\text{Var}(Z) = \text{Var}(X) + \text{Var}(Y)

Worked Examples

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Example 1: PGF of the Sum of Independent Poisson Random Variables

Problem

Suppose $X \sim \text{Po}(\lambda_1)$ and $Y \sim \text{Po}(\lambda_2)$ are independent.

Find the PGF of $Z=X+Y$

Step 1: Write the PGFs of $X$ and $Y$

The PGF for a Poisson random variable $X \sim \text{Po}(\lambda)$ is:

G_X(t) = e^{\lambda(t-1)}

For $X \sim \text{Po}(\lambda_1)$ and $Y \sim \text{Po}(\lambda_2)$ , the PGFs are:

G_X(t) = e^{\lambda_1(t-1)}, \quad G_Y(t) = e^{\lambda_2(t-1)}

Step 2: Find the PGF of $Z=X+Y$

Since $Z=X+Y$ and $X$ and $Y$ are independent, the PGF of $Z$ is:

G_Z(t) = G_X(t) \times G_Y(t)

Substitute:

G_Z(t) = e^{\lambda_1(t-1)} \times e^{\lambda_2(t-1)}

Simplify:

G_Z(t) = e^{(\lambda_1 + \lambda_2)(t-1)}

Final Answer:

The PGF of $Z$ is:

G_Z(t) = e^{\lambda(t-1)}, \quad \text{where } \lambda = \lambda_1 + \lambda_2

This shows that $Z \sim \text{Po}(\lambda_1 + \lambda_2)$

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Example 2: Mean and Variance of the Sum of Binomial Random Variables

Problem

Suppose $X \sim B(5, 0.4)$ and $Y \sim B(7, 0.4)$ are independent.

Find the mean and variance of $Z=X+Y$

Step 1: Write the PGFs of $X$ and $Y$

The PGF of a binomial random variable $X \sim B(n, p)$ is:

G_X(t) = [pt + (1-p)]^n

For $X \sim B(5, 0.4)$ and $Y \sim B(7, 0.4)$ , the PGFs are:

G_X(t) = [0.4t + 0.6]^5, \quad G_Y(t) = [0.4t + 0.6]^7

Step 2: Find the PGF of $Z=X+Y$

Since $Z=X+Y$ and $X$ and $Y$ are independent, the PGF of $Z$ is:

G_Z(t) = G_X(t) \times G_Y(t)

Substitute:

G_Z(t) = [0.4t + 0.6]^5 \times [0.4t + 0.6]^7 = [0.4t + 0.6]^{12}

Step 3: Find the Mean and Variance

For $Z \sim B(12, 0.4)$

Mean: $\mathbb{E}[Z] = np = 12 \times 0.4 = :highlight[4.8]$
Variance: $\text{Var}(Z) = np(1-p) = 12 \times 0.4 \times 0.6 = :highlight[2.88]$

Final Answer:

Mean: $\mathbb{E}[Z] = :highlight[4.8]$
Variance: $\text{Var}(Z) = :highlight[2.88]$

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Example 3: Transformation of a Geometric Random Variable

Problem

Suppose $X \sim \text{Geom}(0.5)$

Let $Z = 2X$ . Find the PGF of $Z$ .

Step 1: Write the PGF of $X$

For $X \sim \text{Geom}(p)$ , the PGF is:

G_X(t) = \frac{p}{1-(1-p)t}, \quad |t| < \frac{1}{1-p}

Substitute $p = 0.5$

G_X(t) = \frac{0.5}{1-0.5t}

Step 2: Apply the Scaling Transformation

For $Z = 2X$ , the PGF of $Z$ is:

G_Z(t) = G_X(t^2)

Substitute $G_X(t)$

G_Z(t) = \frac{0.5}{1-0.5t^2}

Final Answer:

The PGF of $Z$ is:

G_Z(t) = \frac{0.5}{1-0.5t^2}

Note Summary

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Common Mistakes

Forgetting independence: The product of PGFs applies only to independent random variables.
Incorrect application of transformations: Ensure the correct form is used for scaling or shifts.
Neglecting parameter changes: For sums, verify whether distribution parameters (e.g., $\lambda$ or $n$ ) are additive.

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Key Formulas

Sum of Independent Random Variables:

G_Z(t) = G_X(t) \times G_Y(t)

Scaling Transformation:

G_Z(t) = G_X(t^a), \quad \text{for } Z = aXe

Shifted Sum Transformation:

G_Z(t) = t^c G_X(t), \quad \text{for } Z = X + c

Probability Generating Functions (Edexcel A-Level Further Mathematics): Revision Notes

20.1.3 PGFs of Sums & Transformations

Introduction

PGFs of Sums of Independent Random Variables

Transformations of Random Variables

Mean and Variance of Sums

Mean of $Z = X + Y$

Variance of $Z = X + Y$ (independent $X$ and $Y$ )

Worked Examples

Example 1: PGF of the Sum of Independent Poisson Random Variables

Example 2: Mean and Variance of the Sum of Binomial Random Variables

Example 3: Transformation of a Geometric Random Variable

Note Summary

Common Mistakes

Key Formulas

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