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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

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Conditional Densities Example: Suppose X and Y are jointly continuous with joint density f (x, y) = 2 0 for 0 ≤ x ≤ y ≤ 1, otherwise.

Determine the conditional density of X given Y for each value of y.

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y

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x

If we know that Y = 3/4, what is the probability that X ≤ 1/2?

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Conditional Densities Example: Suppose X and Y are jointly continuous with joint density f (x, y) = e−(x+y) 0 for x ≥ 0, y ≥ 0, otherwise.

Determine the conditional density of Y given X for each value of x.

x

y

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Conditional Densities Observation: Let X and Y be jointly continuous random variables having joint density f (x, y) and marginal densities f1(x) and f2(y). Consider the conditional density f (x|y0) of X given Y . It satisﬁes ◦ for each y0, f (x|y0) ≥ 0 for every x;

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