Consider two distributions P1x and P2x over X. Let P1x be a uniform distribution and P2x be any other distribution. Can we say that
Variance of P1 >= Variance of P2
2026-04-08 19:38:53.1775677133
Variance of Uniform Distribution vs Any other Distribution
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No, this is not the case. Consider $X = [0,1]$, then the uniform distribution has variance $1/12$ while the distribution $P_2(0) = P_2(1) = 1/2$ has variance $1/4$.