Question

Let X 1 ,X 2 ,…,X n be n independent random variables. They all have equal mean μ 0 and equal variance σ 02

​ What are the mean μ Y and variance σ Y2 of Y= n1 ∑ i=1n (Xi −μ 0 )
(A) μ Y =0,σ Y2 =σ 02
(B) μ Y =μ 0 ,σ Y2 =σ 02 /n
(C) μ Y =μ 0 ⋅σ Y2 =σ 02
​(D) μv=0⋅σ v2 =σ 02 / n
​

Answer 1

The combined mean (μY) and variance (σY^2) of the sum of n independent random variables with equal mean (μ0) and variance (σ_0^2) are 0 and σ_0^2 respectively, given that Y represents the sum of the deviations from μ0 for each variable.

Step-by-step explanation:

The student is asking about the combined mean (μY) and variance (σY2) of the sum of n independent random variables that have equal mean (μ0) and equal variance (σ02). To answer this, we should recognize that if X1, X2, …, Xn are independent, then the sum of their deviations from the mean (μ0) would be:

Y = ∑i=1n (Xi - μ0)

The mean (μY) for Y would be the sum of the means of the individual variables minus nμ0, which is 0 since each individual mean is μ0. The variance (σY2) for Y would be n times the variance of each variable because variances add up for independent variables, so it would be nσ02. However, since we're looking at the sum of deviations, each term's variance is σ02, so adding n of these gives nσ02, which simplifies back to σ02 since we're talking about deviations.

The correct answer is (A) μY = 0, σY2 = σ02.