Introduction to Linear Transformations of Random Variables

A <strong>linear transformation</strong> of a random variable is one of the most powerful tools in probability and statistics — it lets us rescale, shift, and combine random variables while keeping our calculations tractable.

Linear Transformation of a Random Variable

For a random variable $X$ , a linear transformation takes the form $a X + b$ , where $a$ and $b$ are real constants. The constant $a$ is a scale factor and $b$ is a shift (translation).

Linear transformations appear constantly in real-world contexts:

Converting temperatures from Celsius to Fahrenheit: $F = 1.8 C + 32$
Converting units (e.g., cm to inches)
Applying a fixed markup or tax to a random price

The key results for a single random variable $X$ are:

$E (a X + b) = a E (X) + b$
$Var (a X + b) = a^{2} Var (X)$

These two formulas form the foundation of this subtopic. Understanding why they work, and how to apply them carefully, is essential for HL exam success.

Analogy

Think of $X$ as a factory's daily output (random). If you multiply every outcome by $a$ (change the unit price) and add $b$ (a fixed daily overhead), the average output scales by $a$ and shifts by $b$ — but the spread only changes with $a$ , not $b$ , because $b$ shifts everything equally.

Expected Value of a Linear Transformation

The expected value (mean) of a linear transformation of a random variable $X$ satisfies:

$E (a X + b) = a E (X) + b$

This result follows directly from the linearity of expectation — one of the most important principles in probability theory.

Proof sketch (discrete case):

$E (a X + b) = \sum_{x} (a x + b) P (X = x) = a \sum_{x} x P (X = x) + b \sum_{x} P (X = x) = a E (X) + b$

Since all probabilities sum to 1, the $b$ term simply contributes $b \cdot 1 = b$ .

Note

This formula is valid for both discrete and continuous random variables. The linearity of expectation does not require independence.

Example

Unit conversion example:

Suppose the height $X$ of a randomly selected student (in cm) has $E (X) = 170$ cm. We convert to inches using the transformation $Y = 0.3937 X$ .

$E (Y) = E (0.3937 X) = 0.3937 \times 170 = 66.93 inches$

Now suppose there is also a fixed platform of height $2$ inches, so $Y = 0.3937 X + 2$ :

$E (Y) = 0.3937 \times 170 + 2 = 66.93 + 2 = 68.93 inches$

Warning

A common error is to write $E (a X + b) = a E (X)$ and forget the constant $b$ . The shift $b$ does affect the expected value — it shifts the entire distribution.

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AHL 4.14—Linear transformation of a single RV, E(X) and VAR(X), unbiased estimators

Introduction to Linear Transformations of Random Variables

Expected Value of a Linear Transformation

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