notes8 - MTH4106 Introduction to Statistics Notes 8 Spring...

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Unformatted text preview: MTH4106 Introduction to Statistics Notes 8 Spring 2012 Two or more random variables Two continuous random variables If X and Y are continuous random variables defined on the same sample space, they have a joint probability density function f X , Y ( x , y ) such that, if A is any region of R 2 , then P (( X , Y ) ∈ A ) = Z Z ( x , y ) ∈ A f X , Y ( x , y ) d x d y . Since probabilities are non-negative, we have f X , Y ( x , y ) > 0 for all real x and y . Putting A = R 2 gives Z ∞- ∞ Z ∞- ∞ f X , Y ( x , y ) d x d y = 1 . It can also be shown that if δ x and δ y are small and positive then P ( x 6 X 6 x + δ x and y 6 Y 6 y + δ y ) ≈ f X , Y ( x , y ) δ x δ y . x x + δ x y y + δ y . . . . . . . . . . . . . . . . . . . . . . . . . . The joint cumulative distribution function F X , Y is given by F X , Y ( x , y ) = P ( X 6 x and Y 6 y ) = Z y- ∞ Z x- ∞ f X , Y ( t , u ) d t d u . 1 The marginal probability density functions are obtained by integrating over the other variable: f X ( x ) = Z ∞- ∞ f X , Y ( x , y ) d y ; f Y ( y ) = Z ∞- ∞ f X , Y ( x , y ) d x ....
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notes8 - MTH4106 Introduction to Statistics Notes 8 Spring...

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