If the random variable V∼N(μ,σ2),σ2>0, then the random variable W=σ2(X−μ)2=Z2∼χ2(1), chi-squared distribution, with ν=1 degrees of freedom.
Suppose X∼N(0,1), then Y=12(X−0)2=X2∼χ2(1), chi-squared distribution, with ν=1 degrees of freedom.
Its density function is given by
f(y;ν)=⎩⎨⎧2ν/2Γ(ν/2)1yν/2−1e−y/2,y>00,elsewhere Then
f(y;ν)=⎩⎨⎧2πy1e−y/2,y>00,elsewhere The mean of the distribution is equal to the number of degrees of freedom:
E(Y)=μY=ν=1 The variance is equal to two times the number of degrees of freedom:
Var(Y)=σY2=2ν=2
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