Mathematics (9709) | Linear combinations of random variables

Linear Combinations of Random Variables (Paper 6, Section 6.2)

Hello! Welcome to one of the most practical and frequently tested topics in P2 Statistics: Linear Combinations of Random Variables. Don't worry about the long name—this chapter is all about figuring out the mean and variance (the expected value and the spread) when you combine, add, or subtract different random outcomes.

Think of it like this: If you know the average time and variability for your morning commute (X) and your afternoon return trip (Y), how do you find the average and variability of your total daily travel time (X + Y)? This chapter gives you the simple rules to do exactly that!

1. Transforming a Single Random Variable: \(aX + b\)

Before we combine two variables (like \(X+Y\)), we first look at transforming a single variable \(X\). This is often necessary when calculating costs, scaling measurements, or converting units.

A. Rules for Expectation (The Mean)

The expected value, or mean (\(E\)), measures the centre of the distribution. It follows very intuitive rules:

Rule 1: Scaling and Shifting

\[ E(aX + b) = aE(X) + b \]

Analogy: Imagine \(X\) is the price of a coffee with \(E(X) = \$4\). If the government adds a fixed tax of \(\$0.50\) (\(b=0.5\)), and then the shop doubles the base price (\(a=2\)), the new expected price is simply \(2(\$4) + \$0.50 = \$8.50\). Expectation is easy—it’s linear.

B. Rules for Variance (The Spread)

The variance (\(Var\)) measures the spread or variability around the mean. This is where students often make small errors, so pay close attention!

Rule 2: Variance is Unaffected by Shifting

Adding or subtracting a constant (\(b\)) to every value shifts the entire distribution, but it does not change how spread out the values are.

\[ Var(X + b) = Var(X) \]

Rule 3: Variance Squares the Scaling Factor

If you multiply the variable by a constant \(a\), the spread is multiplied by \(a^2\). This is because variance is calculated using squared differences from the mean.

\[ Var(aX + b) = a^2 Var(X) \]

Key Point: The constant \(b\) vanishes entirely in the variance calculation.

2. Combining Two Independent Random Variables: \(aX \pm bY\)

In the real world, we rarely deal with just one variable. We need rules for combining two (or more) different variables, like finding the total weight of two randomly chosen apples (\(X+Y\)).

The crucial condition for the variance rule below is that the variables \(X\) and \(Y\) must be independent. (In AS & A Level Maths, you will generally assume independence unless told otherwise.)

A. Rules for Combined Expectation

Expectation remains beautifully straightforward and always combines linearly:

Rule 4: Expectation of Sums and Differences

The mean of the sum is the sum of the means. The mean of the difference is the difference of the means.

\[ E(aX + bY) = aE(X) + bE(Y) \]

Note: This rule works even if X and Y are not independent, but the syllabus focuses on its application alongside the independence assumption for variance.

B. The Golden Rule for Combined Variance

This is the most critical and often misused formula in this chapter.

Rule 5: Variances Always Add (For Independent Variables)

Whether you are calculating the variance of a sum (\(X+Y\)) or the variance of a difference (\(X-Y\)), the individual variances ALWAYS ADD.

\[ Var(aX \pm bY) = Var(aX) + Var(bY) = a^2 Var(X) + b^2 Var(Y) \]

Why do variances add, even for subtraction?
Analogy: Consider two journey times, X and Y. If you take the total time \(X+Y\), the uncertainty increases. If you find the difference \(X-Y\), the uncertainty still increases! You are uncertain about X and uncertain about Y. Subtracting them doesn't magically cancel the uncertainty; it compounds it. You are less certain about the final outcome, meaning the spread (variance) is larger.

Common Mistake Alert!
Never write \(Var(X - Y) = Var(X) - Var(Y)\). This is incorrect. If you see a subtraction sign between the variables, immediately switch your brain to addition for variance: \(Var(X - Y) = Var(X) + Var(Y)\).

3. Combining Specific Distributions

When combining random variables, the shape of the resulting distribution matters, especially if you plan to calculate probabilities (e.g., using Normal distribution tables). The key concept here is that for certain common distributions, the result of a linear combination often keeps the same distributional family.

A. Linear Combinations of Normal Variables

If \(X\) is normally distributed, and \(Y\) is normally distributed, their linear combination is always normally distributed (provided they are independent).

If \(X \sim N(\mu_X, \sigma_X^2)\) and \(Y \sim N(\mu_Y, \sigma_Y^2)\), and they are independent, then:

\[ aX \pm bY \sim N \left( E(aX \pm bY), \ Var(aX \pm bY) \right) \]

What this means: If you are adding or subtracting two Normal variables, the new variable is also Normal. You just need to calculate the new mean and the new variance using the rules from Section 2.

Example Scenario: The weight of an adult male (\(M\)) is Normal, and the weight of a child (\(C\)) is Normal. If you select one of each, the total weight (\(M+C\)) is also Normal. This is vital because it means you can use the standard Normal distribution (\(Z\)) to find probabilities for the total weight.

B. Linear Combinations of Poisson Variables

The syllabus requires you to know the result for the sum of two independent Poisson variables. Scaling or subtracting Poisson variables is generally outside the scope of Paper 6.

Rule 6: Sum of Independent Poisson Variables

If \(X\) and \(Y\) are independent Poisson distributions, their sum \(X+Y\) is also a Poisson distribution, where the mean parameters (\(\lambda\)) are added.

If \(X \sim Po(\lambda_X)\) and \(Y \sim Po(\lambda_Y)\), and they are independent, then:

\[ X + Y \sim Po(\lambda_X + \lambda_Y) \]

Example Scenario: The number of emails received by Jane (\(X\)) follows \(Po(2)\) and the number received by David (\(Y\)) follows \(Po(3)\). If they are independent, the total number of emails received by both (\(X+Y\)) follows \(Po(2+3) = Po(5)\).

This property is consistent with the general rules, as for a Poisson distribution, \(E(X) = \lambda\) and \(Var(X) = \lambda\).

• \(E(X+Y) = E(X) + E(Y) = \lambda_X + \lambda_Y\)
• \(Var(X+Y) = Var(X) + Var(Y) = \lambda_X + \lambda_Y\)

Since the new variable \(X+Y\) has a mean and variance of \(\lambda_X + \lambda_Y\), it fits the criteria for a Poisson distribution with parameter \(\lambda_X + \lambda_Y\).

4. Step-by-Step Problem Solving Strategy

Most problems involving linear combinations require a careful application of the expectation and variance rules before calculating a probability.

Example: Total Production Time

The time (in minutes) taken to assemble component A is \(T_A \sim N(10, 4)\). The time taken for component B is \(T_B \sim N(15, 9)\). A worker completes 3 component A's and 2 component B's independently. Find the distribution of the total time, \(T_{Total}\).

Step 1: Define the combined variable.
The total time is \(T_{Total} = T_{A1} + T_{A2} + T_{A3} + T_{B1} + T_{B2}\).
Alternatively, since all \(T_A\) are independent and identical, we can write: \[ T_{Total} = 3T_A + 2T_B \]

Step 2: Calculate the Expected Value (\(E\)).

\[ E(T_{Total}) = E(3T_A + 2T_B) = 3E(T_A) + 2E(T_B) \] \[ E(T_{Total}) = 3(10) + 2(15) = 30 + 30 = 60 \text{ minutes} \]

Step 3: Calculate the Variance (\(Var\)).

Remember, variances always add, and coefficients are squared!

\[ Var(T_{Total}) = Var(3T_A + 2T_B) = 3^2 Var(T_A) + 2^2 Var(T_B) \] \[ Var(T_{Total}) = 9(4) + 4(9) = 36 + 36 = 72 \]

Step 4: State the resulting distribution.

Since both \(T_A\) and \(T_B\) are Normal, their combination is also Normal.

\[ T_{Total} \sim N(60, 72) \]

You would then use this new distribution to solve any probability questions, using the standard Normal Z-transformation.

Important Note on the Squared Coefficient:
A frequent source of confusion is the difference between \(X_1 + X_2 + X_3\) and \(3X\).

• If you are looking at the sum of three independent observations (like three different randomly selected components), the variance is \(Var(X_1) + Var(X_2) + Var(X_3)\). If they are identical, this is \(3 Var(X)\).
• If you are looking at one observation multiplied by 3 (a scaling factor), the variance is \(Var(3X) = 3^2 Var(X) = 9 Var(X)\).

Make sure your formula reflects the reality of the question: are you summing multiple items, or scaling one item? The formula \(Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)\) handles both cases elegantly when \(Y\) is just a constant zero variable.