Sample Variance

Var(Xn)=S2n=∑ni=1(Xi−X¯n)2n−1

E(S2n)=E⎛⎝⎜⎜∑ni=1(Xi−X¯n)2n−1⎞⎠⎟⎟

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(n−1)E(S2n)n−1nE(Sn)2======⇔=E(∑i=1n(X1−X¯n)2)E(∑i=1nX2i)−E(∑i=1n2XiX¯n)+E(∑i=1nX¯2n)E(∑i=1nX2i)−E(2X¯n∑i=1nXi)+E(X¯2n∑i=1n1)E(∑i=1nX2i)−E(2nX¯n)2+E(nX¯2n)E(∑i=1nX2i)−nE(X¯2n)nE(X2i)−nE(X¯2n)E(X2i)−E(X¯2n)

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E(X2i)==E(Xi)2+V(X)μ2+σ2

E(X¯2n)====E(X¯n)2+V(X¯n)μ2+V(1n∑i=1nXi)μ2+1n2V(∑ni=1Xi)μ2+σ2n

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n−1nE(Sn)2==μ2+σ2−μ2−σ2nn−1nσ2

E(Sn)2=σ2