What's the variance of the average of n points of random variables X.

All X's are independent and identically distributed.

That is, what's V(
¯
X
), for i = 1 to n?

____________________________________________________

(reminder: )

Question

What's the variance of the average of n points of  random variables X.

All X's are independent and identically distributed.

That is, what's V(
¯
X
), for i = 1 to n?

____________________________________________________

(reminder:  )

BlackTom AI · Accepted Answer

Let's break down the problem carefully and proceed step by step.
First, we are given a set of random variables X1, X2, ..., Xn that are independent and identically distributed (i.i.d.). The quantity of interest is the variance of the sample mean, denoted X̄ = (X1 + X2 + ... + Xn) / n.
Now, consider how the variance of a sum behaves when variables are independent. For independent variables, Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) and cross-terms vanish because Cov(X, Y) = 0. Extending this to the sum X1 + X2 + ... + Xn, we have Var(X1 + X2 + ... + Xn) = Var(X1) + Var(X2) + ... + Var(Xn) = n Var(X) since all Vi are identical with Var(X).
Next, the sample mean X̄ is the sum divided by n, i.e., X̄ = (1/n) (X1 + X2 + ... + Xn). Using the property Var(aY) = a^2 Var(Y) for a constant a, we get Var(X̄) = (1/n^2) Var(X1 + X2 + ... + Xn) = (1/n^2) * n Var(X) = Var(X) / n.
So the variance of the average of n i.i.d. random variables is Var(X)/n.
Now, evaluating the provided options (or the implied correct option if the list is incomplete):
- If the option states 1/n * Var(X) (i.e., Var(X)/n), that matches the derived result exactly. This is the correct form because it reflects the division by n due to averaging, while preserving the original variability of X via Var(X).
- If there were any options suggesting Var(X)/n^2, that would be incorrect because the variance of the sum scales with n, and the division by n twice would be inappropriate for the mean.
- If an option claimed Var(X)/√n or Var(X)*n or any nonstandard combination, those would conflict with the linearity and independence properties used above and would be incorrect.
- In summary, the correct expression is Var(X)/n, which corresponds to the standard result Var(X̄) = Var(X)/n for i.i.d. samples.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 8. Quiz for Class 8

What's the variance of the average of n points of random variables X. All X's are independent and identically distributed. That is, what's V( ¯ X ), for i = 1 to n? ____________________________________________________ (reminder: )

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