What is the average variance of the sum of independent random variables according to the central limit theorem?

What is the average variance of the sum of independent random variables according to the central limit theorem?

• A. nσ^2
• B. σ^2/n
• C. σ
• D. nμ

Answer: (A)

The average variance of the sum of independent random variables is represented by the formula for the variance of the sum. When you have \( n \) independent random variables, each with the same variance \( \sigma^2 \), the variance of their sum is simply the sum of their individual variances. This leads to the expression \( n\sigma^2 \), which indicates that the total variance increases linearly with the number of variables being summed. In the context of the central limit theorem, as the number of independent random variables increases, their sum tends toward a normal distribution, regardless of the original distribution of the variables. This property is particularly useful when assessing the distribution of averages and sums in probabilistic models. The other options do not accurately reflect the behavior of the variance in this scenario. For instance, \( σ^2/n \) would suggest a diminishing variance as you increase the number of random variables, which is not the case for sums. Similarly, \( σ \) and \( nμ \) relate to standard deviation and the expected value, respectively, but do not pertain to the variance of the sum as described by the central limit theorem. Thus, recognizing that the average variance of the sum of independent random variables is \( n

The average variance of the sum of independent random variables is represented by the formula for the variance of the sum. When you have ( n ) independent random variables, each with the same variance ( \sigma^2 ), the variance of their sum is simply the sum of their individual variances. This leads to the expression ( n\sigma^2 ), which indicates that the total variance increases linearly with the number of variables being summed.

In the context of the central limit theorem, as the number of independent random variables increases, their sum tends toward a normal distribution, regardless of the original distribution of the variables. This property is particularly useful when assessing the distribution of averages and sums in probabilistic models.

The other options do not accurately reflect the behavior of the variance in this scenario. For instance, ( σ^2/n ) would suggest a diminishing variance as you increase the number of random variables, which is not the case for sums. Similarly, ( σ ) and ( nμ ) relate to standard deviation and the expected value, respectively, but do not pertain to the variance of the sum as described by the central limit theorem. Thus, recognizing that the average variance of the sum of independent random variables is ( n