Introduction

In the realm of probability theory and statistics, the concept of independent and identically distributed (iid) random variables plays a crucial role in simplifying calculations and understanding the behavior of random variables in various scenarios. This chapter delves into the significance of the iid property when computing the mean and variance of a sum of iid random variables. We’ll explore how this property brings about simplifications and insights that are invaluable in statistical analysis.

IID Property

Independent and identically distributed (iid) random variables are variables that are mutually independent and have the same probability distribution. This property simplifies calculations because it implies that the behavior of each variable is unaffected by the others and they all share the same distribution characteristics. This makes them a powerful tool in various statistical calculations.

Computing the Mean and Variance of a Sum

• Mean: When considering the sum of $n$ iid random variables $X_1, X_2, \ldots, X_n$, the mean of the sum is simply $n$ times the mean of any one of the random variables:

$$E\left(X_1+X_2+\ldots+X_n\right)=n \times E\left(X_1\right)$$

• Variance: The variance of the sum of iid random variables is the sum of their individual variances:

$$\operatorname{Var}\left(X_1+X_2+\ldots+X_n\right)=\operatorname{Var}\left(X_1\right)+\operatorname{Var}\left(X_2\right)+\ldots+\operatorname{Var}\left(X_n\right)$$

Example: Consider the toss of a fair six-sided die. Let $X$ be the result of a single toss. Since the die is fair, each outcome is equally likely, and the mean of $X$ is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5. Now, let $Y$ be the result of another independent toss. Since the two tosses are independent and fair, the mean of $Y$ is also 3.5. If we want to find the mean of the sum $X + Y$, we simply add the means: $E(X + Y)$ = $E(X)$ + $E(Y)$ = 3.5 + 3.5 = 7.