Probabilities for linear combinations of random variables

Welcome to the World of Linear Combinations!

Hello future actuaries! So far in your Exam P journey, you’ve likely spent a lot of time looking at single random variables. But in the real world—and especially in insurance—we rarely look at just one thing at a time. An insurance company doesn't just care about one policy; they care about the total sum of thousands of policies. That’s where linear combinations come in!

In this chapter, we are going to learn how to calculate the mean, variance, and probabilities when we add, subtract, or scale random variables. Don't worry if this seems a bit abstract at first; once you see the patterns, it becomes one of the most "points-rich" areas of the exam!

What exactly is a Linear Combination?

Think of a linear combination as a "math recipe." If you have two random variables, \( X \) (say, the claim amount for car accidents) and \( Y \) (the claim amount for home fires), a linear combination is simply a new variable \( W \) created by multiplying them by constants and adding them together.

The general form looks like this:
\( W = aX + bY + c \)

Where \( a \), \( b \), and \( c \) are just regular numbers (constants).
• \( a \) and \( b \) are the weights (scaling factors).
• \( c \) is a fixed shift (like a flat processing fee added to every claim).

A Quick Analogy

Imagine you are buying 2 pounds of apples (\( X \)) and 3 pounds of bananas (\( Y \)). If the price per pound is a random variable because it fluctuates daily, your total cost is \( 2X + 3Y \). That is a linear combination!

Step 1: Finding the Expected Value (The "Easy" Part)

The Expected Value (or Mean) is very well-behaved. It follows the rule of Linearity. This is a fancy way of saying you can just distribute the \( E[\cdot] \) symbol to every piece of the equation.

The Formula:
\( E[aX + bY + c] = aE[X] + bE[Y] + c \)

Why this is great: You do not need to know if \( X \) and \( Y \) are independent to use this formula. It works every single time! If you are adding two variables, you just add their averages. Simple as that.

Quick Review: The Rules of E

• \( E[cX] = cE[X] \)
• \( E[X + Y] = E[X] + E[Y] \)
• \( E[c] = c \) (The average of a constant is just that constant!)

Step 2: Finding the Variance (The "Tricky" Part)

Variance is a bit more sensitive than the mean. It doesn't just add up cleanly because variables might "interact" with each other. This interaction is measured by Covariance.

The Formula:
\( Var(aX + bY + c) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y) \)

Important Note: Notice that the constant \( c \) disappears! Variance measures spread, and shifting a distribution by a constant amount doesn't change how "spread out" it is. Also, notice that the coefficients \( a \) and \( b \) are squared.

Wait, what is Covariance?

Covariance \( Cov(X, Y) \) tells us if the variables move together.
• If \( Cov(X, Y) > 0 \): When \( X \) goes up, \( Y \) tends to go up.
• If \( Cov(X, Y) < 0 \): When \( X \) goes up, \( Y \) tends to go down.
• If \( Cov(X, Y) = 0 \): There is no linear relationship.

Did you know? If \( X \) and \( Y \) are independent, then \( Cov(X, Y) = 0 \). This makes the formula much shorter:
\( Var(aX + bY) = a^2Var(X) + b^2Var(Y) \)

Common Pitfall: The Minus Sign

Students often make mistakes when subtracting variables. If you are finding \( Var(X - Y) \), the formula is:
\( Var(X - Y) = Var(X) + Var(Y) - 2Cov(X, Y) \)
If they are independent, it becomes:
\( Var(X - Y) = Var(X) + Var(Y) \)
Never subtract variances! Variance is a measure of uncertainty; if you combine two uncertain things, your total uncertainty usually goes up, not down.

Step 3: Calculating Probabilities

Now that we have the new Mean and new Variance, how do we find actual probabilities (like \( P(X + Y > 10) \))?

Method A: The Normal Distribution (Most Common)

If the problem tells you that \( X \) and \( Y \) are normally distributed, then any linear combination of them is also normally distributed. This is a massive shortcut!

Step-by-Step Process:
1. Find the new mean: \( \mu_W = E[aX + bY] \).
2. Find the new variance: \( \sigma^2_W = Var(aX + bY) \).
3. Take the square root to get the new standard deviation: \( \sigma_W \).
4. Use the Z-score formula: \( Z = \frac{W - \mu_W}{\sigma_W} \).
5. Look up the probability in the standard normal table.

Method B: The Moment Generating Function (MGF) Trick

If the variables aren't normal, we sometimes use MGFs. The MGF of a sum of independent variables is the product of their individual MGFs.
\( M_{X+Y}(t) = M_X(t) \cdot M_Y(t) \)

If you recognize the resulting MGF as belonging to a specific distribution (like Gamma or Poisson), you can solve the probability using that distribution's properties.

Summary & Key Takeaways

1. Mean is easy: Just distribute the expectation. Independence doesn't matter.
2. Variance needs care: Remember to square the coefficients and include the \( 2abCov(X, Y) \) term unless you are told they are independent.
3. Covariance is key: \( Cov(X, Y) = E[XY] - E[X]E[Y] \).
4. Normal stays Normal: Sums of normal variables are normal. This is your "best friend" for many Exam P questions.
5. Watch the units: If a question asks for the standard deviation of the sum, don't forget to take the square root of your final variance calculation!

Don't worry if the covariance part feels heavy! Just remember: Independence = 0 Covariance. Most exam questions will either give you the covariance or tell you the variables are independent to simplify your life. You've got this!