Mean and Variance of Squared Gaussian
Question
Given Y=X2 where X∼ϕ(0,σ2), what is the mean and variance of the Y?
Since σ2=E(X2)−E(X)2, and E(X)=0, E(X2)=σ2. Also,
VarX2=EX4−(EX2)2 and the fourth moment EX4 is equal to 3σ4 since
∫2πσx4e−2σ2x2dx=2πσ1(−σ2x3e−2σ2x2∣−∞+∞+3σ2∫x2e−2σ2x2dx)=2πσ1(0−3σ4xe−2σ2x2∣−∞+∞+3σ4∫e−2σ2x2dx)=3σ4 (actually, E[X2n]=(2n−1)!!σ2n, !! is the double factorial
thus
VarX2=EX4−(EX2)2=3σ4−σ4=2σ4 Reference
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