Variance of a sum

Definition

If $X_1, X_2, \dots, X_n$ are random variables, then the variance of their sum is:

$$\mathrm{Var}(X_1 + X_2 + \cdots + X_n)$$

This quantity measures how much the total $X_1 + X_2 + \cdots + X_n$ varies around its mean.

For two random variables $X$ and $Y$,

$$\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)+2\,\mathrm{Cov}(X,Y)$$

where $\mathrm{Cov}(X,Y)$ is the covariance between $X$ and $Y$.

If $X$ and $Y$ are independent, then $\mathrm{Cov}(X,Y)=0$, so:

$$\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)$$

Main Content

1. First Concept: Variance of a Sum for Two Random Variables

• The variance of a sum is not found by simply adding the numbers unless the variables are independent.
• For two random variables $X$ and $Y$, the full rule is:

$$\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)+2\,\mathrm{Cov}(X,Y)$$

This formula shows three parts:

1. the variation from $X$,
2. the variation from $Y$,
3. the interaction between them, measured by covariance.

If covariance is positive, the variables tend to increase together, making the variance of the sum larger. If covariance is negative, one tends to increase when the other decreases, which can reduce the total variance.

Example:
Suppose $\mathrm{Var}(X)=4$, $\mathrm{Var}(Y)=9$, and $\mathrm{Cov}(X,Y)=3$.

Then:

$$\mathrm{Var}(X+Y)=4+9+2(3)=19$$

If instead $\mathrm{Cov}(X,Y)=0$, then:

$$\mathrm{Var}(X+Y)=4+9=13$$

This clearly shows how dependence changes the spread of the sum.

2. Second Concept: Independence and Additivity of Variance

• When random variables are independent, their covariance is zero.
• Therefore, for independent variables, the variance of the sum is just the sum of the variances:

$$\mathrm{Var}(X_1+X_2+\cdots+X_n)=\mathrm{Var}(X_1)+\mathrm{Var}(X_2)+\cdots+\mathrm{Var}(X_n)$$

This is a very powerful result because independence makes calculations much simpler.

If $X_1, X_2, \dots, X_n$ are independent and each has variance $\sigma^2$, then:

$$\mathrm{Var}(X_1+\cdots+X_n)=n\sigma^2$$

So if you add more independent random variables, the total variability increases linearly with the number of variables.

Example:
Let $X$ and $Y$ be independent with variances $5$ and $8$.

Then:

$$\mathrm{Var}(X+Y)=5+8=13$$

This property is widely used in sampling, repeated measurements, and probability distributions.

3. Third Concept: General Formula Using Covariances

• For many random variables, the variance of a sum includes all pairwise covariance terms.
• The general formula is:

$$\mathrm{Var}\left(\sum_{i=1}^{n}X_i\right) = \sum_{i=1}^{n}\mathrm{Var}(X_i) + 2\sum_{1\le i<j\le n}\mathrm{Cov}(X_i,X_j)$$

This means:

• add all individual variances,
• then add twice every covariance between distinct pairs.

If the variables are mutually independent, all covariance terms become zero, and the formula reduces to the sum of variances.

For three variables $X, Y, Z$:

$$\mathrm{Var}(X+Y+Z)=\mathrm{Var}(X)+\mathrm{Var}(Y)+\mathrm{Var}(Z) +2\mathrm{Cov}(X,Y)+2\mathrm{Cov}(X,Z)+2\mathrm{Cov}(Y,Z)$$

Example:
Suppose:

• $\mathrm{Var}(X)=2$
• $\mathrm{Var}(Y)=3$
• $\mathrm{Var}(Z)=4$
• $\mathrm{Cov}(X,Y)=1$
• $\mathrm{Cov}(X,Z)=0$
• $\mathrm{Cov}(Y,Z)=-2$

Then:

$$\mathrm{Var}(X+Y+Z)=2+3+4+2(1)+2(0)+2(-2)=9+2-4=7$$

This shows how both positive and negative relationships affect the total spread.

Working / Process

1. Identify the random variables being added
   Determine whether you are finding the variance of $X+Y$, $X+Y+Z$, or a larger sum. Write the sum clearly before starting calculations.

2. Check whether the variables are independent
   If they are independent, the problem becomes simpler because all covariance terms are zero. If they are not independent, you must include covariance terms.

3. Apply the correct variance formula

4. For two variables:
   $$\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)+2\mathrm{Cov}(X,Y)$$

5. For many variables:
   $$\mathrm{Var}\left(\sum_{i=1}^{n}X_i\right) = \sum_{i=1}^{n}\mathrm{Var}(X_i) + 2\sum_{i<j}\mathrm{Cov}(X_i,X_j)$$

Visual idea for the process:

Total Sum
   |
   v
Check independence?
   |---- yes ----> Add variances only
   |
   |---- no -----> Add variances + covariance terms

Example process:
Find $\mathrm{Var}(A+B)$ when $A$ and $B$ are independent and $\mathrm{Var}(A)=6$, $\mathrm{Var}(B)=10$.

Step 1: Write the sum $A+B$.
Step 2: Since they are independent, covariance is $0$.
Step 3: Add variances:

$$\mathrm{Var}(A+B)=6+10=16$$

Advantages / Applications

• Helps in analyzing the spread of combined random outcomes, such as total marks, total sales, total production, or total waiting time.
• Useful in statistics and data analysis for understanding how repeated measurements and sample totals behave.
• Important in probability, finance, engineering, and science for measuring risk, error accumulation, and uncertainty propagation.

Summary

• The variance of a sum measures the spread of a total random quantity.
• For independent random variables, variances simply add.
• For dependent variables, covariance terms must also be included.
• Important terms to remember: variance, covariance, independence.