For a random variable x, the central moments (\(\mu \)_i) of all order exist. The square of (2j + 1)^th moment (\(\mu^2_2{_j}{_+}{_1}\)) is

Question

Question

This question was previously asked in

SSC CGL JSO Tier-II, 2018 (Statistics) Official Paper-III (Held On: 14 Sept, 2019)

Answer 1

Explanation

For a random variable, central moment(μ) of all order exist

Square of (2j + 1)^th moment (μ²_{2j + 1})

Let j = 1 then (2j + 1)^th moment is given by

(μ²₃) = [(∑(x – x̅)³/N]²

⇒ (∑x³/N)² ------(i)

If j = 1 then μ₂ = ∑x²/N

⇒ μ_{2j + 2} if j = 1 then

⇒ μ₄ = ∑x⁴/N

⇒ μ₂ × μ_{2j + 2} = ∑x²/N × ∑x⁴/N ------(ii)

Compare both equation

(∑x³)²/N² = (∑x² × ∑x⁴)/N²

⇒ (∑x³)² ≤ (∑x² × ∑x⁴)

∴ (μ_{2j + 1})² ≤ μ_2j × μ_2j_{+ 2}