Joint Probability Distributions Example: Let X and Y be jointly continuous random variables having joint density f (x, y) = 3y 0 for 0 ≤ x ≤ y ≤ 1, otherwise.

Sketch the domain of f as well as f .
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Determine P (Y ≤ 2X).

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Jointly Discrete Probability Distributions Definition 5.1: Let X and Y be discrete random variables. The joint probability mass function of X and Y is p(x, y) = P (X = x, Y = y) which is defined for all x and y. Remark: The joint distribution can be given in tabular form. For example: y 0 1 2 0 1/9 2/9 1/9 x 1 2/9 2/9 0 2 1/9 0 0

Definition 5.4: The marginal mass functions of X and Y , respectively, are p1(x) = y p(x, y) and p2(y) = x p(x, y).

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Jointly Discrete Probability Distributions Definition 5.5: If X and Y are jointly discrete random variables with joint mass function p and marginal mass functions p1 and p2, then the conditional mass function of X given Y is p(x, y) p(x|y) = P (X = x|Y = y) = p2(y) provided P (y) > 0.

Example: Determine the conditional mass function p(x|y) for the joint distribution given by y 0 1 2 0 1/9 2/9 1/9 x 1 2/9 2/9 0 2 1/9 0 0

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Jointly Continuous Probability Distributions Definition 5.4: If X and Y are jointly continuous random variables with joint density f (x, y), then the marginal density functions of X and Y , respectively, are
∞ ∞

f1(x) =
−∞

f (x, y) dy

and f2(y) =
−∞

f (x, y) dx.

Example: Let X and Y be jointly continuous random variables having joint density f (x, y) = 3x 0 for 0 ≤ y ≤ x ≤ 1, otherwise.

Determine the marginal densities f1 and f2.

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Conditional Densities Definition 5.7: Let X and Y be jointly continuous random variables with joint density f and marginal densities f1 and f2. Then the conditional density of X given Y is   f (x, y)  when f2(y) > 0, f2(y) f (x|y) =   0 when f2(y) = 0 and the conditional