... Thefunctions x and y arerequired toberandom variables. Note that z : S →∗×∗, and that we need z−1([−∞,α] ×[−∞,β]) ∈ for all real α and β.The two-dimensional mapping performed by the ... 4.5: PMF and CDF representations for Example 4.1.3.(a) Sketch the two-dimensional representations for the PMF and the CDF. (b) Find px. (c) Find py.(d) Find P(x < y). (e) Are x and y independent?Solution. ... Dx× Dy and satisfyingpx,y(α, β) ≥ 0 for all real α and β, (4.14)px(α) =β∈Dypx,y(α, β), (4.15) and py(β) =α∈Dxpx,y(α, β), (4.16)where px and pyare valid one-dimensional...