Question

Let ₁,₂,…,ₙ be independently and identically distributed random variables with nonzero variance. Then the variance of₁+…+ₙ is equal to the variance of ₁.

A. True
B. False

Answer 1

The variance of the sum of independent and identically distributed random variables is not equal to the variance of a single random variable but is n times larger, making the statement False.

Step-by-step explanation:

When independent and identically distributed random variables X1, X2, ..., Xn with nonzero variance are summed together, the variance of the sum ΣXi (that is the variance of X1 + X2 + ... + Xn) is not equal to the variance of a single random variable X1. This is because, by the properties of variance for independent random variables, the variance of the sum is the sum of the variances of the individual random variables. Therefore, the variance of the sum is n times the variance of one of the random variables if they are identically distributed. Hence, the statement is False.