The bookUnit 4 · Probability73 / 83

Joint distributions, transformations, order statistics

Why this is asked: Uncorrelated is weaker than independent except for the joint normal. Use the Jacobian formula for transformations and the standard order-statistic densities.

Joint distributions, transformations and order statistics

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Order statistics from first principlesinteractive

Derive the max, the min and the k-th — with the dependence that catches people out.

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The trap here

“If X and Y are each normal and uncorrelated then they are independent” — false

X ~ $N(0,1), \varepsilon = \pm1$ with probability ½ independent of $X, Y = \varepsilon{}X$

Y is N(0,1), Cov(X,Y) = 0, but |X| = |Y| always. The pair is not jointly normal.