Stat & Thm

Question 1

Sample Mean (x̄)

Answer 1

Sn/n = ∑Xi/n, i = 1,…,n

Question 2

Sample Variance (S²)

Answer 2

(∑Xi - x̄)²/(n-1), i = 1,…,n

Question 3

rth sample raw moments (M_r’)

Answer 3

∑Xi^r/n, i = 1,…,n, ∀r = 1,2,3,…

Question 4

sample central moment (M_r)

Answer 4

(∑Xi - x̄)^r/n, i = 1,…,n, ∀r = 1,2,3,…

Question 5

order stat

Answer 5

X_(1) ≤ X_(2) ≤ … ≤ X_(n)

Question 6

Let x̄ = (X1, X2,…,Xn)’ be a r.s. fr Fx, then ∀r = 1,2,3,… E(Mr’) & V(Mr’)

Answer 6

E(Mr’) = E(X^r) = μ^r if E(X^r) exists

V(Mr’) = {E[X^2r] - [E(X^r)]^2}/n

Question 7

Let x̄ = (X1, X2,…,Xn)’ be a r.s. fr Fx with μ & σ² (Exp & Var of Sample Var)

Answer 7

E(S²) = σ²

V(S²) = [μ_4 - {(n-3)σ^4/n-1}]/n

Question 8

∑(Xi - x̄)²

Answer 8

∑(Xi - μ)² - n(x̄ - μ)²

Question 9

∑(Xi - μ)²

Answer 9

∑(Xi - x̄)² + n(x̄ - μ)²