Math 408 Actuarial Statistics I A J Hildebrand Joint Distributions Continuous Case In the following X and Y are continuous random variables Most of the concepts and formulas below are analogous to those for the discrete case with integrals replacing sums The principal difference between continuous lies in the definition of the p d f p m f f x y The formula f x y P X x Y y is no longer valid and there is no simple and direct way to obtain f x y from X and Y 1 Joint continuous distributions RR Joint density joint p d f A function f x y satisfying i f x y 0 ii f x y dxdy 1 Usually f x y will be given by an explicit formula along with a range a region in the xy plane on which this formula holds In the general formulas below if a range of integration is not explicitly given the integrals are to be taken over the range in which the density function is defined Uniform joint distribution An important special type of joint density is one that is constant over a given range a region in the xy plane and 0 outside outside this range the constant being the reciprocal of the area of of the range This is analogous to the concept of an ordinary one variable uniform density f x over an interval I which is constant and equal to the reciprocal of the length of I inside the interval and 0 outside it 2 Marginal distributions The ordinary distributions of X and Y when considered separately The corresponding one variable densities are denoted by fX or f1 and fY or f2 and obtained by integrating the joint density f x y over the other variable Z Z fX x f x y dy fY y f x y dx 3 Computations with joint distributions Probabilities Given a region R in the xy plane the probability that X Y falls into this region is given by the double integral of fRR x y over this region For example P X Y 1 is given by an integral of the form R f x y dxdy where R consists of the part of the range of f in which x y 1 Expectation RR of a function of X and Y e g u x y xy E u X Y u x y f x y dxdy 4 Covariance and correlation The formulas and definitions are the same as in the discrete case Definitions Cov X Y E XY E X E Y E X X Y Y Covariance Correlation of X and Y of X and Y X Y Cov X Y X Y Properties Cov X Y X Y 1 X Y 1 Relation to variance Var X Cov X X Variance of a sum Var X Y Var X Var Y 2 Cov X Y 5 Independence of random variables Same as in the discrete case Definition X and Y are called independent if the joint p d f is the product of the individual p d f s i e if f x y fX x fY y for all x y 1 Math 408 Actuarial Statistics I A J Hildebrand Properties of independent random variables If X and Y are independent then The expectation of the product of X and Y is the product of the individual expectations E XY E X E Y More generally this product formula holds for any expectation of a function X times a function of Y For example E X 2 Y 3 E X 2 E Y 3 The product formula holds for probabilities of the form P some condition on X some condition on Y where the comma denotes and For example P X 2 Y 3 P X 2 P Y 3 The covariance and correlation of X and Y are 0 Cov X Y 0 X Y 0 The variance of the sum of X and Y is the sum of the individual variances Var X Y Var X Var Y The moment generating function of the sum of X and Y is the product of the individual moment generating functions MX Y t MX t MY t 6 Conditional distributions Same as in the discrete case with integrals in place of sums Definitions conditional density of X given that Y y g x y ff x y Y y conditional density of Y given that X x x y h y x ffX x Conditional expectations and variance Conditional expectations variances etc are defined and computed as usual but with conditional distributions in place of ordinary distributions For example R E X Y 1 E X Y 1 xg x 1 dx R E X 2 Y 1 E X 2 Y 1 x2 g x 1 dx 2
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