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Lecture

# Joint distributions handout.pdf

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School
Department
Statistics
Course
STAT 330
Professor
Christine Dupont
Semester
Fall

Description
Joint Distributions Suppose X and Y are deﬁned on the same sample space S. 1. Joint c.d.f.: F(x;y) = P(X ▯ x;Y ▯ y) 2. Properties of F: (a) For ﬁxed x, F is non-decreasing in y (b) For ﬁxed y, F is non-decreasing in x (c) lim F(x;y) = 0 and lim F(x;y) = 0 x!▯1 y!▯1 (d) lim F(x;y) = 0and lim F(x;y) = 1 (x;y)!(▯1;▯1) (x;y)!(1;1) 3. (a) Marginal c.d.f. of X:x) = P(X ▯ x) = y!1 F(x;y) = F(x;1) (b) Marginal c.d.f. of Y:y) = P(Y ▯ y) = lim F(x;y) = F(1;y) x!1 4. Joint Discrete Random Variables (Suppose the sample space S is discrete): (a) Joint p.f.: f(x;y) = P(X = x;Y = y). Support set A = f(x;y) : f(x;y) > 0g (b) Properties of f: 2 i. f(x;y) ▯ 0 8(x;y) 2 R X X ii. lim F(x;y) = f(x;y) = 1 (x;y)!(1;1) (x;y)A
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