Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
5.D.3. Joint Probability Distributions Let X and Y be stochastic variables on S with values X S x1, x2 , Y S y1, y2 , The product set X S Y S x , y , x , y , , x , y , 1 1 1 2 i is a probability space if every ordered pair j x , y i j is assigned a probability P X xi , Y y j f xi , y j where f xi , y j is called the joint probability distribution of X and Y. The switch to continuous stochastic variables is easily accomplished by making use of the joint probability distribution density f x, y defined by P x X x x, y Y y y f x, y xy and replacing all sums with integrals. By definition, f obeys the usual properties of a probability density, namely, 1. Positive-definiteness: f x, y 0 . 2. Normalization: dxdy f x, y 1 . Making use of the addition principle (see §5.B), we have f X x dyf x, y fY y dxf x, y (5.22) The covariance of X and Y is defined by cov X , Y dxdy x X y Y f x, y dxdy xy x Y y X X Y XY X Y Y X X Y XY X Y (5.24) The correlation of X and Y is defined by cov X , Y corr X , Y (5.25) XY X XY X Y X 2 f x, y 2 Y 2 Y 2 It is dimensionless with the following easily proved properties: corr X ,Y corr Y , X . 1. Symmetric: 2. corr X , X 1 , 3. 1 corr X ,Y 1 . 4. corr aX b, cY d corr X ,Y corr X , X 1 . if a, c 0 . Note that pairs of stochastic variables with identical distributions can have different correlations if the joint distributions are different. If X and Y are statistically independent, we have 1. f x, y f X x fY y . 2. XY X Y . 3. X Y 4. 2 X Y 2 X2 X 2 Y2 Y 2 . cov X , Y 0 . Note that cov X , Y 0 does not imply X and Y are statistically independent. Consider the variable z G x, y where G is a known function of x and y. The probability density f Z z for the stochastic variable Z is given by f Z z dxdy z G x, y f x, y (5.26) Using x 1 dke ikx 2 we have fZ z 1 2 dkdxdye ik z G x , y f x, y which, on comparison with fZ z 1 dke ikzZ k 2 gives Z k dxdyeikG x , y f x, y (5.28a) If X and Y are statistically independent, i.e., f x, y f X x fY y we have Z k dxdyeikG x , y f X x fY y (5.28) For example, if G x, y x y then Z k dxdyeik x y f X x fY y X k Y k (5.29) i.e., the characteristic function of the sum of 2 independent stochastic variables is the product of their characteristic functions.