F Y (y) ＝ F (+ , y) ＝ ＝ P{Y  y} 3.2 Marginal distribution F X (x) ＝ F (x, +  ) ＝ ＝ P{X  x} Marginal distribution function for bivariate Define –P57 the marginal cdfs of ( X, Y) with respect to X and Y respectively

Example 1. Suppose that the joint distribution of (X,Y) is specified by Determine F X (x) and F Y (y) 。

Marginal distribution for discrete distribution Suppose that (X, Y) ～ P{X ＝ x i, Y ＝ y j,} ＝ p ij ， i, j ＝ 1, 2, … Defin e-P57 P{X ＝ x i } ＝ p i. ＝ ， i ＝ 1, 2, … P{Y ＝ y j } ＝ p. j ＝ ， j ＝ 1, 2, … the marginal pmf of (X, Y) with respect to X and Y respectively.

Marginal density function the marginal pdf of (X,Y) with respect to X and Y. Example 3.4-P59 Suppose that (X, Y) ～ f (x, y), (x, y)  R 2, define

the marginal pdf of Y

Example 3. Suppose that the joint density function of (X,Y)is specified by Determine (1) the value of c; (2)the marginal distribution of (X,Y) with respect to X

(1) Bivariate uniform distribution Bivariate (X, Y) is said to follow uniform distribution if the density function is specified by By the definition, one can easily to find that if (X, Y) is uniformly distributed, then

Suppose that (X,Y) is uniformly distributed on area D, which is indicated by the graph on the right hand, try to determine: (1)the density function of (X,Y) ； (2)P{Y<2X} ； (3)F(0.5,0.5) Answer

where ，  1 、  2 are constants and  1 >0,  2 >0 、 |  |<1 are also constant, then, it is said that (X, Y) follows the two-dimensional normal distribution with parameters  1,  2,  1,  2,  and denoted it by (2)Two dimensional normal distribution Suppose that the density function of (X, Y) is specified by

Example The joint pdf of (X,Y) is Find the marginal pdf of X and Y.

Solution temporary fix when so temporary fix

when So

Homework : P67: 5,6