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 claims across 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 real constants:
• \( a \) and \( b \) are the weights or scaling factors.
• \( c \) is a fixed shift (like a flat administration fee added to claims).

A Quick Analogy

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

Step 1: Finding the Expected Value (The Linear Part)

The Expected Value (or mean) is very well-behaved. It follows the rule of linearity of expectation. You distribute the expectation operator to each term:

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

Why this is great: You do not need \( X \) and \( Y \) to be independent to use this formula. It holds for all random variables with finite expectations.

Quick Review: Rules of Expectation

• \( E[aX] = aE[X] \)
• \( E[X + Y] = E[X] + E[Y] \)
• \( E[c] = c \) (The expected value of a constant is just that constant)

Step 2: Finding the Variance

Variance measures spread, so adding a constant does not change the variance. When combining variables, we must account for their relationship using Covariance.

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

Important Note: Notice that \( c \) vanishes, and the coefficients \( a \) and \( b \) are squared.

What is Covariance?

Covariance \( Cov(X, Y) = E[XY] - E[X]E[Y] \) measures the linear association between \( X \) and \( Y \). If \( X \) and \( Y \) are independent, then \( Cov(X, Y) = 0 \). Under independence, the variance simplifies to:
\( Var(aX + bY) = a^2Var(X) + b^2Var(Y) \)

Common Pitfall: Subtracting Variables

When finding the variance of a difference \( X - Y \), the coefficient of \( Y \) is \( -1 \). Since \( (-1)^2 = 1 \):
\( Var(X - Y) = Var(X) + Var(Y) - 2Cov(X, Y) \)
If \( X \) and \( Y \) are independent:
\( Var(X - Y) = Var(X) + Var(Y) \)
Never subtract variances! Uncertainties add together.

Step 3: Calculating Probabilities for Linear Combinations

Exam P focuses on calculating probabilities for linear combinations in two primary settings: continuous normal random variables and independent discrete random variables, as well as large-sample approximations via the Central Limit Theorem.

Setting 1: Continuous Normal Random Variables

Any linear combination of independent normal random variables is also normally distributed. If \( X \sim N(\mu_X, \sigma_X^2) \) and \( Y \sim N(\mu_Y, \sigma_Y^2) \) are independent, and \( W = aX + bY \), then:
• Mean: \( \mu_W = a\mu_X + b\mu_Y \)
• Variance: \( \sigma_W^2 = a^2\sigma_X^2 + b^2\sigma_Y^2 \)
• Distribution: \( W \sim N(\mu_W, \sigma_W^2) \)

To find a probability such as \( P(W \le w) \):
1. Calculate the standard deviation: \( \sigma_W = \sqrt{\sigma_W^2} \).
2. Standardize: \( Z = \frac{W - \mu_W}{\sigma_W} \).
3. Look up \( P(Z \le z) = \Phi(z) \) in the standard normal distribution table provided on the exam.

Setting 2: Independent Discrete Random Variables

When \( X \) and \( Y \) are discrete and independent, the probability mass function of \( W = aX + bY \) is computed by summing the joint probabilities over all pairs \( (x, y) \) that satisfy the condition.

Because \( X \) and \( Y \) are independent, the joint probability factors: \( P(X = x, Y = y) = P(X = x)P(Y = y) \).

Discrete Convolution Formula:
For the sum \( S = X + Y \):
\( P(S = s) = \sum_{x} P(X = x) P(Y = s - x) \)

Example: Suppose independent claim counts \( X \) and \( Y \) have probability functions \( P(X = 0) = 0.6 \), \( P(X = 1) = 0.4 \), and \( P(Y = 0) = 0.7 \), \( P(Y = 1) = 0.3 \). To find \( P(X + Y = 1) \):
\( P(X + Y = 1) = P(X = 0, Y = 1) + P(X = 1, Y = 0) = (0.6)(0.3) + (0.4)(0.7) = 0.18 + 0.28 = 0.46 \).

Special Discrete Families:
• If \( X \sim \text{Poisson}(\lambda_1) \) and \( Y \sim \text{Poisson}(\lambda_2) \) are independent, then \( X + Y \sim \text{Poisson}(\lambda_1 + \lambda_2) \).
• If \( X \sim \text{Binomial}(n_1, p) \) and \( Y \sim \text{Binomial}(n_2, p) \) are independent with the same success probability \( p \), then \( X + Y \sim \text{Binomial}(n_1 + n_2, p) \).

Setting 3: Central Limit Theorem (Approximating Probabilities)

When dealing with a linear combination of a large number of independent and identically distributed (i.i.d.) random variables, \( S_n = \sum_{i=1}^n X_i \), the exact distribution may be complex, but the Central Limit Theorem (CLT) allows a normal approximation:

\( E[S_n] = n\mu \), \( \quad Var(S_n) = n\sigma^2 \)

For sufficiently large \( n \), the standardized sum is approximately standard normal:
\( Z = \frac{S_n - n\mu}{\sigma\sqrt{n}} \approx N(0, 1) \)

Summary & Key Takeaways

1. Mean is linear: Always distribute expectation: \( E[aX + bY] = aE[X] + bE[Y] \), regardless of independence.
2. Variance requires covariance: \( Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y) \). If independent, the covariance term is 0.
3. Normal stays Normal: Linear combinations of independent normal variables remain normal, allowing direct standardization into \( Z \).
4. Discrete sums use convolution: For independent discrete variables, sum the products \( P(X = x)P(Y = y) \) over all pairs yielding the desired value.
5. CLT for large sums: Approximate the sum of i.i.d. variables using a normal distribution with mean \( n\mu \) and standard deviation \( \sigma\sqrt{n} \).