Properties of Mean and Variance Revision Notes for VCE SSCE Mathematical Methods

Properties of Mean and Variance

Introduction to expected values and linear functions

When working with continuous random variables, it's important to understand that the expected value of a function of $X$ is not necessarily equal to that function applied to the expected value of $X$ . In mathematical terms:

E[g(X)] \neq g[E(X)]

This means we cannot simply apply a function to the mean and expect to get the mean of the transformed variable.

infoNote

However, there is an important exception to this rule: linear functions. When the function $g$ is linear (of the form $aX + b$ ), the mean of the linear function equals the linear function of the mean. This special property makes linear transformations much easier to work with.

The mean of $aX + b$

For any continuous random variable $X$ , the expected value of a linear transformation follows this formula:

E(aX + b) = aE(X) + b

This tells us that:

Multiplying $X$ by $a$ multiplies the mean by $a$
Adding $b$ to $X$ adds $b$ to the mean

Proof of the mean formula

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Proof: Expected Value of Linear Transformation

We can prove this result using the definition of expected value and properties of integrals:

E(aX + b) = \int_{-\infty}^{\infty} (ax + b)f(x) \, dx

= \int_{-\infty}^{\infty} ax f(x) \, dx + \int_{-\infty}^{\infty} b f(x) \, dx

= a \int_{-\infty}^{\infty} x f(x) \, dx + b \int_{-\infty}^{\infty} f(x) \, dx

= aE(X) + b

The last step uses the fact that $\int_{-\infty}^{\infty} f(x) \, dx = 1$ (since $f$ is a probability density function).

The variance of $aX + b$

For any continuous random variable $X$ , the variance of a linear transformation follows this formula:

\text{Var}(aX + b) = a^2 \text{Var}(X)

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Notice that the constant $b$ does not appear in this formula. This makes intuitive sense: adding a constant merely shifts the probability density function horizontally without changing its spread. The variance measures spread, so translation doesn't affect it.

Proof of the variance formula

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Proof: Variance of Linear Transformation

We start with the definition of variance and use our result for expected values:

\text{Var}(aX + b) = E[(aX + b)^2] - [E(aX + b)]^2

First, let's expand $[E(aX + b)]^2$ :

[E(aX + b)]^2 = [aE(X) + b]^2 = (a\mu + b)^2 = a^2\mu^2 + 2ab\mu + b^2

where $\mu = E(X)$ .

Next, let's find $E[(aX + b)^2]$ :

E[(aX + b)^2] = E(a^2X^2 + 2abX + b^2)

= a^2E(X^2) + 2ab\mu + b^2

Now we can substitute into the variance formula:

\text{Var}(aX + b) = a^2E(X^2) + 2ab\mu + b^2 - a^2\mu^2 - 2ab\mu - b^2

= a^2E(X^2) - a^2\mu^2

= a^2[E(X^2) - \mu^2]

= a^2 \text{Var}(X)

Worked example

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Worked Example: Finding Mean and Variance of Linear Transformations

Suppose that $X$ is a continuous random variable with mean $\mu = 10$ and variance $\sigma^2 = 2$ .

a) Find $E(2X + 1)$

E(2X + 1) = 2E(X) + 1

= 2 \times 10 + 1

= 21

b) Find $\text{Var}(1 - 3X)$

\text{Var}(1 - 3X) = (-3)^2 \text{Var}(X)

= 9 \times 2

= 18

Note that we write $1 - 3X$ as $-3X + 1$ , so $a = -3$ and we square this value.

The probability density function of $aX + b$

When we transform a random variable, we also need to understand how its probability density function changes. Linear transformations affect both the shape and position of the PDF.

The random variable $X + b$

If the probability density function of $X$ has rule $f(x)$ , then the probability density function of $X + b$ is obtained by a horizontal translation. The new PDF has rule:

f(x - b)

This represents a translation of $b$ units in the positive direction along the $x$ -axis (assuming $b > 0$ ).

The random variable $aX$

Multiplying by a constant $a$ is similar to a dilation from the y-axis. However, we need to adjust the rule to preserve the total area under the curve (which must remain equal to $1$ ). The probability density function of $aX$ has rule:

\frac{1}{a} f\left(\frac{x}{a}\right)

The factor $\frac{1}{a}$ ensures the area under the transformed curve remains $1$ .

The random variable $aX + b$

Combining both transformations, if the probability density function of $X$ has rule $f(x)$ , then the probability density function of $aX + b$ has rule:

\frac{1}{a} f\left(\frac{x - b}{a}\right)

This transformation can be described by the mapping:

(x, y) \to \left(ax + b, \frac{y}{a}\right)

When $a$ and $b$ are positive, this represents:

A dilation of factor $a$ from the $y$ -axis
A dilation of factor $\frac{1}{a}$ from the $x$ -axis
A translation of $b$ units in the positive direction along the $x$ -axis

Worked example

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Worked Example: Transforming a Probability Density Function

Suppose that $X$ is a continuous random variable with probability density function:

f(x) = \begin{cases} \frac{2(x - 1)}{25} & 1 \leq x \leq 6 \\ 0 & \text{elsewhere} \end{cases}

Find the probability density function $g$ for $2X + 3$ .

Solution

The transformation is described by $(x, y) \to \left(2x + 3, \frac{y}{2}\right)$ .

For points on the $x$ -axis: $(x, 0) \to (2x + 3, 0)$ for all $x \in \mathbb{R}$ .

The endpoints of the domain transform as follows:

$(1, 0) \to (5, 0)$
$\left(6, \frac{2}{5}\right) \to \left(15, \frac{1}{5}\right)$

The graph with rule $y = \frac{2(x - 1)}{25}$ is mapped to the graph with rule:

y = \frac{1}{2} \times \frac{2\left(\frac{x - 3}{2} - 1\right)}{25}

= \frac{1}{2} \times \frac{2(x - 5)}{50}

= \frac{x - 5}{50}

Therefore:

g(x) = \begin{cases} \frac{x - 5}{50} & 5 \leq x \leq 15 \\ 0 & \text{elsewhere} \end{cases}

chatImportant

Exam Tip: Always check that your transformed domain makes sense by applying the transformation to the original endpoints.

Remember!

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Key Points to Remember:

For linear functions, the expected value distributes: $E(aX + b) = aE(X) + b$
The variance of a linear transformation is: $\text{Var}(aX + b) = a^2\text{Var}(X)$
Adding a constant $b$ does not affect variance - it only shifts the distribution without changing its spread
When transforming a PDF, use the rule $\frac{1}{a}f\left(\frac{x-b}{a}\right)$ for $aX + b$
Always verify that the transformed PDF integrates to $1$ over the new domain

Properties of Mean and Variance (VCE SSCE Mathematical Methods): Revision Notes