Chapter 5a - Chapter 5 Joint Densities Discrete Variables...

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Chapter 5 Joint Densities Discrete Variables Let X and Y be random variables. The ordered pair (X,Y) is called a two-dimensional discrete random variable. A function f XY such that f XY (x,y) = P( X=x and Y=y) is called the joint density for (X,Y). Conditions for a Discrete Joint Density 1) f XY (x,y) > 0 2) XY f (x,y) = 1 X Y ∑∑ Example: Show this is a valid joint density function. Y=1 Y=2 X=1 .1 .3 X=2 .4 .2
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Discrete Marginal Density Let (X,Y) be a two-dimensional discrete random variable with joint density f XY . The marginal density for X, denoted by f X is given by ( ) ( , ) x XY Y f x f x y = The marginal density for Y, denoted by f y , is given by ( ) ( , ) y XY X f y f x y = Expectation of Discrete Joint Densities [ ] ( , ) XY X Y E X xf x y = ∑∑ ∑∑ = X Y y x f x x E ) , ( ) ( 2 2 σ x 2 = E(x 2 ) – [E(x)] 2 [ ] ( , ) XY X Y E Y yf x y = ∑∑ ∑∑ = X Y y x f y y E ) , ( ) ( 2 2 σ Y 2 = E(y 2 ) – [E(y)] 2
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(a) Find the marginal density for X. (b) Find the mean and standard deviation for X.
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This note was uploaded on 01/17/2011 for the course STAT 4714 at Virginia Tech.

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Chapter 5a - Chapter 5 Joint Densities Discrete Variables...

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