Question

4. suppose we have independent observations x1 and x2, both from a distribution with expected value µ and standard deviation σ. what is the variance of the average of the two values:

Answer 1

The variance of the average of two independent observations from a distribution with standard deviation σ is σ²/2.

Step-by-step explanation:

The question is asking for the variance of the average of two independent observations, x1 and x2, from a distribution with known expected value μ and standard deviation σ. To calculate the variance of the average, we use the properties of the variance of independent random variables.

The formula for the variance of the average is:
Var((x1 + x2)/2) = (1/4)[Var(x1) + Var(x2)]
Since both x1 and x2 are from the same distribution and are independent, Var(x1) = Var(x2) = σ². Substituting these values in the formula, we get:
Var((x1 + x2)/2) = (1/4)[σ² + σ²] = (1/4)[2σ²] = σ²/2.

Therefore, the variance of the average of two independent observations from a distribution with standard deviation σ is σ²/2.

Answer 2

The variance of the average of two independent observations is equal to the variance of the original distribution.

Step-by-step explanation:

To find the variance of the average of two independent observations, we can use the formula for the variance of the mean:

Variance of the average = (Variance of x1 + Variance of x2) / 2

Since x1 and x2 are both from a distribution with expected value µ and standard deviation σ, we can substitute their respective variances:

Variance of the average = (σ + σ²) / 2 = 2σ² / ² = σ²

Therefore, the variance of the average of the two values is equal to the variance of the original distribution, which is σ².