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Let X ∼ Unif[−1, 1]. Consider the functions g, h : [−1, 1] → [−1, 1]...

Let X ∼ Unif[−1, 1]. Consider the functions g, h : [−1, 1] → [−1, 1] given by g(x) = 1 − x if x ∈ [0, 1]; x if x ∈ [−1, 0), and h(x) = x if x ∈ [0, 1]; −(x + 1) if x ∈ [−1, 0).

Y = g(X) and Z = h(X) are both uniform Y, Z ∼ Unif[−1, 1].

(c) Prove that the random vectors (X, Y ) and (X, Z) do not have the same joint distribution. This can be done by finding a subset B ⊂ R 2 such that P((X, Y ) ∈ B) 6= P((X, Z) ∈ B).

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