Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Remarks: Support(X,Y)is the set of all points (x,y)in the xy plane such that the probability that (X,Y)takes values in a small neighborhood of each of these points is always strictly positive. Since fxy(x,y)=0 for all points (x,y)outside the sup- port of (X,Y),calculation of probabilities for any events related to (X,Y)will only involve integrals of fxy(x,y) on the support xy. Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 36/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 36/370 Random Vectors and Joint Probability Distributions Remarks: The Continuous Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Example 3(5.3) Suppose X and Y have a joint PDF fxy(x,y)=cy2if0≤x≤2and0≤y≤1， where c is an unknown constant. Find (1)the value of c; (2)P(X+Y>2): (3)P(X<0.5)月 (4)P(X=3Y). Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 37/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 37/370 Random Vectors and Joint Probability Distributions The Continuous Case Example 3 (5.3)
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Solution The support2xy={(x,y):0≤x≤2,0≤y≤1}isa rectangular area in the xy plane. (L)Recalling∫ofxY(c,y）drdy=l,we have (-( dy 2c dy 2c 3 2c 3 Thus,c= To be Continued Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 38/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 38/370 Random Vectors and Joint Probability Distributions The Continuous Case Solution To be Continued
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case (2) x-了>y=(,) 3-8 (3) 1 (4)Because x =3y is a line,the volume over a line is zero.Therefore,we have P(X=3Y)=0. Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 39/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 39/370 Random Vectors and Joint Probability Distributions The Continuous Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Example 4(5.4) Suppose X and Y have a joint PDF fxy(x,y)=cx2 y for a2≤y≤1， where c is an unknown constant. Find (1)the value of c; (2)P(X≥Y). Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 40/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 40/370 Random Vectors and Joint Probability Distributions The Continuous Case Example 4 (5.4)