Bivariate or joint probability distributions

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4 Example 3.4 The random variables X and Y have a bivariate normal distribution if ( ) − f x, y = ae b where a= 2π σ σ 1− ρ X 1 Y 2 and b = 1 2 ( − ρ ) 2 1 ⎡ ⎛ x − µ ⎞ ⎛ X x − µ ⎞ ⎛ X y − µ ⎞ ⎛ Y y − µ ⎢ ⎜ ⎟ − 2ρ⎜ ⎟ ⎜ ⎟ + ⎜ ⎣⎢ ⎝ σ X ⎠ ⎝ σ X ⎠ ⎝ σ Y ⎠ ⎝ σ Y 2 2 Y ⎞ ⎟ ⎠ ⎤ ⎥ ⎦⎥ where −∞< x
3.9 Conditional probability density functions The conditional p.d.f of X given Y = y is denoted by f ( x y) and defined as f ( x y ) = f ( x Y y) = = 5 ( , ) ( y) f x y whereas the conditional p.d.f of Y given X = x is denoted by f ( y x) and defined as f ( y x ) = f ( y X x) = = f Y ( , ) ( x) f x y f X 3.10 Joint probability distribution function As in 3.5 the joint (cumulative) probability distribution function (c.d.f.) is denoted by F(x, y) and is defined as F(x, y) = P( X ó x and Y ó y) but F(x, y) in the continuous case is the volume under the p.d.f. surface from X = −∞ to X = x and from Y = −∞ to Y = y, so that F( x y) v= y ∫ u= x , = ( , ) v=−∞ ∫ u=−∞ f u v du dv The marginal c.d.f. ‘s are defined as in 3.5 and can be obtained from the joint distribution function F(x, y) as follows: F F X ( x) = F( x y MAX ) Y ( y) = F( x y) , where y MAX is the largest value of y and MAX , where x MAX is the largest value of x. 3.11 Important connections between the p.d.f ‘s and the joint c.d.f.’s. (i) The joinf p.d.f. f (x, y) = ( , ) ∂ 2 F x y ∂x∂y (ii) The marginal p.d.f’s can be obtained from the marginal c.d.f.’s as follows: the marginal p.d.f. of X = f ( x) X = the marginal p.d.f. of Y = f ( y) Y = dF X dx dF y Y dy ( x) ( ) or F ′ ( x) X , or F ′ ( y) Y 3.12 Are X and Y independent? X and Y are independent random variables if either (a) F(x, y) = FX ( x) FY ( y) ; or
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