Moments for linear combinations of random variables

Exam P – Probability · SOA (Society of Actuaries) · 5 min read

Welcome to the World of Linear Combinations!

Welcome! Today, we are going to learn how to handle "teams" of random variables. In the previous chapters, you spent a lot of time looking at single variables (like the cost of one insurance claim). But in the real world—and on Exam P—we often deal with a mix of variables.

Think of it like managing a portfolio: you don't just care about one stock; you care about the total value of all your stocks combined. This chapter teaches you the mathematical "shortcuts" to find the Mean (Expected Value) and Variance of these combined groups without having to do a mountain of calculus.

Don't worry if this seems tricky at first; once you see the patterns, it’s actually one of the most "points-friendly" sections of the exam!

1. The Superpower of Expectation: Linearity

The Expected Value (or Mean) is incredibly friendly. It follows a rule called Linearity.

If you have two random variables, $X$ and $Y$, and some constant numbers $a$, $b$, and $c$, the expected value of their combination is exactly what you would hope it is:

$ E[aX + bY + c] = aE[X] + bE[Y] + c $

Why is this a superpower? Because this rule works regardless of whether $X$ and $Y$ are independent! They could be best friends, bitter enemies, or total strangers; the Expected Value just adds up linearly either way.

Real-World Analogy: The Grocery Trip

Imagine $X$ is the price of a gallon of milk and $Y$ is the price of a loaf of bread. If you buy 2 gallons of milk ($2X$) and 3 loaves of bread ($3Y$), and you have a fixed $ \$1 $ bag fee ($c$), your average total cost is just:\n
\n 2 times the average price of milk + 3 times the average price of bread + the $1 fee.

Quick Review Box:
• $ E[X + Y] = E[X] + E[Y] $
• $ E[aX] = aE[X] $
• $ E[c] = c $ (The "average" of a constant is just that constant!)

2. Covariance: Measuring the Relationship

Before we talk about Variance, we need to meet Covariance. Covariance measures how two variables move together.

The formula for Covariance is:
$ Cov(X, Y) = E[XY] - E[X]E[Y] $

Key Points to Remember:
1. If $X$ and $Y$ are Independent, then $Cov(X, Y) = 0$. (But be careful: the reverse isn't always true!)
2. $Cov(X, X)$ is just a fancy way of saying $Var(X)$.
3. If $Cov(X, Y)$ is positive, they tend to grow together. If it's negative, when one goes up, the other tends to go down.

3. Variance of a Linear Combination: The "FOIL" Method

Variance is a bit more stubborn than Expectation. It doesn't just add up cleanly because variables can interfere with each other.

The formula for the variance of a sum is:
$ Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y) $

How to remember this?

Think back to high school algebra when you squared a binomial: $(m + n)^2 = m^2 + n^2 + 2mn$.
The Variance formula looks almost identical!
• The $a^2$ and $b^2$ terms are the "squares."
• The $2abCov(X, Y)$ is the "middle term" that accounts for how the variables interact.

Common Mistake Alert! Students often forget to square the constants. If you see $Var(3X)$, the answer is $9Var(X)$, not $3Var(X)$.

What if the variables are Independent?

This is the best-case scenario! If $X$ and $Y$ are independent, their Covariance is zero. The formula simplifies to:
$ Var(aX + bY) = a^2Var(X) + b^2Var(Y) $

Did you know? Even if you are subtracting two independent variables, you still add their variances!
$ Var(X - Y) = Var(X) + (-1)^2Var(Y) = Var(X) + Var(Y) $.
Analogy: If you have two sources of uncertainty, subtracting one from the other doesn't make your life more certain—it adds more chaos!

4. The Correlation Coefficient ($\rho$)

Sometimes the exam gives you the Correlation Coefficient ($\rho$) instead of the Covariance. Correlation is just a "standardized" version of covariance that always stays between -1 and 1.

The relationship is:
$ \rho_{XY} = \frac{Cov(X, Y)}{\sigma_X \sigma_Y} $

Or, rearranged for your variance calculations:
$ Cov(X, Y) = \rho_{XY} \cdot \sigma_X \cdot \sigma_Y $
(Where $\sigma$ is the standard deviation, which is $\sqrt{Var}$).

5. Step-by-Step: Solving a Linear Combination Problem

When you face a problem asking for the variance of $aX + bY$, follow these steps:

Step 1: Identify $Var(X)$ and $Var(Y)$.
Step 2: Identify the relationship. Are they independent? If not, find $Cov(X, Y)$ or the correlation $\rho$.
Step 3: Square your constants ($a^2$ and $b^2$).
Step 4: Plug everything into the "Big Formula": $a^2Var(X) + b^2Var(Y) + 2abCov(X, Y)$.
Step 5: Double-check your signs! If you are calculating $Var(X - Y)$, then $b = -1$. This makes the middle term negative: $-2Cov(X, Y)$.

6. Key Takeaways Summary

• Means are easy: Just plug and play. $E[aX + bY + c] = aE[X] + bE[Y] + c$.
• Variances require squaring: Always square the coefficients ($a \rightarrow a^2$).
• The Interaction Term: Don't forget the $2abCov(X, Y)$ unless you are 100% sure the variables are independent.
• Independence is a gift: It makes $Cov(X, Y) = 0$, which simplifies your work significantly.
• Constants have no variance: $Var(X + 5) = Var(X)$. Adding a fixed number shifts the data but doesn't change how "spread out" it is.

Keep practicing these formulas! They are the bread and butter of actuarial math. You've got this!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the formula for the expected value of a linear combination of random variables, $ E[aX + bY + c] $?

$ E[aX + bY + c] = aE[X] + bE[Y] + c $

Does the linearity of expectation property require the random variables to be independent?

No, the property $ E[X + Y] = E[X] + E[Y] $ holds whether the variables are independent or dependent.

What is the expected value of a constant $ c $?

$ E[c] = c $

What is the formula used to calculate the covariance between two random variables $ X $ and $ Y $?

$ Cov(X, Y) = E[XY] - E[X]E[Y] $

What is the covariance of two independent random variables?

If $ X $ and $ Y $ are independent, then $ Cov(X, Y) = 0 $.

What is the general formula for the variance of a linear combination $ aX + bY $?

$ Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y) $

When calculating the variance of $ aX $, how does the constant $ a $ affect the result?

The constant is squared: $ Var(aX) = a^2Var(X) $.

What is the variance of the sum or difference of two independent random variables?

If independent, $ Var(aX ± bY) = a^2Var(X) + b^2Var(Y) $.

How is the correlation coefficient $ ρ_{XY} $ defined?

$ ρ_{XY} = \frac{Cov(X, Y)}{σ_X σ_Y} $, where $ σ $ is the standard deviation.

How do you find the Covariance when given the Correlation Coefficient and standard deviations?

$ Cov(X, Y) = ρ_{XY} · σ_X · σ_Y $

What is the variance of a random variable plus a constant, $ Var(X + c) $?

$ Var(X + c) = Var(X) $

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