Let X ~ N(μ, σ²), then the fourth central moment of X is:

Question

Question

This question was previously asked in

SSC CGL Tier-II (JSO) 2022 Official Paper (Held On: 4 March 2023)

Answer 1

The correct answer is 3σ4

Key Points

The fourth central moment of a normal distribution (also known as kurtosis) is defined as the expectation of the fourth power of the deviations from the mean, divided by the standard deviation to the fourth power.
The kurtosis of a standard normal distribution (μ=0, σ=1) is 3.
However, if we consider a general normal distribution X ~ N(μ, σ²), the fourth central moment (μ4) is given by: μ₄ = E[(X-μ)⁴]
Since the normal distribution is symmetric, this is equivalent to E[X⁴] - 4μE[X³] + 6μ²E[X²] - 4μ³E[X] + μ⁴. We know that for a normal distribution, E[X] = μ, E[X²] = μ² + σ², E[X³] = μ³ + 3μσ², and E[X⁴] = μ⁴ + 4μ³σ² + 6μ²σ⁴ + 3σ⁴.
Substituting these in, we get: μ₄ = μ⁴ + 4μ³σ² + 6μ²σ⁴ + 3σ⁴ - 4μ(μ³ + 3μσ²) + 6μ²(μ² + σ²) - 4μ³μ + μ⁴ = 3σ⁴

Therefore, the fourth central moment of a normal distribution is 3σ⁴.