经管学会网盘光华方向专业概统lecture

1、Lecture 6Bivariate Distributions Marginal Distributions Independence of two random variablesReview: Random Variable Review: Discrete DistributionsX has a discrete distribution if X can take only a finite number of different values x1,.,xk or, at most, an infinite sequence of different values x1,x2,.

2、The probability function (p.f.) of X is defined to be the function f such that for every real number x, f(x)=Pr(X=x).If x is not one of the possible values of X, f(x)=0.We have orIf A is any subset of the real line, Review: Discrete DistributionsReview: Continuous DistributionsX has a continuous dis

3、tribution if there exists a nonnegative function f defined on the real line, s.t. for any subset A of the real line,f is called the probability density function (p.d.f.) of X. Every p.d.f. must satisfy two requirements: The probability density function (pdf) for a continuous random variable is a fun

4、ction which can be integrated to obtain the probability that the random variable takes a value in a given interval. P(a=X=b)=P(X=a)=0 Review: Continuous DistributionsExample: Education and Monthly Personal eSuppose that education level (X) can take values 1=“below college”, 2=“college”, and 3=“above

5、 college”. Suppose that monthly personal e (Y) can take values 1=“=30000”. Suppose that in certain population, the probabilities for different combinations of education level and monthly personal e are given by the table below. Y 1234 10.20.10.060.04 X 20.090.06 0.10.15 3 0.010.030.080.08Bivariate D

6、istributionsThe joint probability distribution of two random variables is called a bivariate distribution.X and Y have a discrete joint distribution.X and Y each has a discrete distribution.If both X and Y have a finite number of possible values, then there will be a only finite number of possible v

7、alues for the pair (X,Y).Otherwise, there will be an infinite number of possible values for the pair (X,Y).The joint p.f. of X and Y is defined to bef(x,y)=Pr(X=x and Y=y).If (x,y) is not one of the possible values of the pair of random variables, then f(x,y)=0.With all pairs of (x,y) for which f(x,

8、y)0,For any subset A of the xy-plane,Example: Education and Monthly Personal eSuppose that in certain population, the probabilities for different combinations of education level and monthly personal e are given by the table below. Y 1234 10.20.10.060.04 X 20.090.06 0.10.15 3 0.010.030.080.08Continuo

9、us Joint DistributionsX and Y have a continuous joint distribution if there exists a nonnegative function f, defined over the xy-plane, s.t. for any subset A of the plane,f is called the joint probability density function of X and Y.Any individual point, or any infinite sequence of points, in the xy

10、-plane has probability 0.Any one-dimensional curve in the xy-plane has probability 0.The probability that (X,Y) will lie on any specified straight line is 0.The probability that (X,Y) will lie on any specified circle is 0.Example: Calculating Probabilities from A Joint p.d.f.Suppose the joint p.d.f.

11、 of X and Y is:Example: Calculating Probabilities from A Joint p.d.f.Suppose the joint p.d.f. of X and Y is:Solution:Example: Determining A Joint p.d.f. by Geometric MethodsA point (X,Y) is selected at random from inside the circle . What is the joint p.d.f. of X and Y? Let S denote the set of point

12、s in the circleThe joint p.d.f. of X and Y is constant over S and is 0 outside S. We must have Bivariate Distribution FunctionThe joint distribution function, or joint d.f., of X and Y is defined to be the function F, s.t. The d.f. F1 of X can be derived from the joint d.f. F as follows, forIf F2 de

13、notes the d.f. of Y, then forIf X and Y have a continuous joint distribution with joint p.d.f. f, then the joint d.f. is The p.d.f. can be derived from the joint d.f.at every point (x,y) where this second-order derivative exists.Example: Determining a Joint p.d.f. from a Joint d.f.The joint d.f. of

14、X and Y, foris . What is the d.f. F1 of X? What is the joint p.d.f of X and Y? Solution: Marginal Distributionsjoint d.f. marginal d.f.joint p.f. or p.d.f marginal p.f. or p.d.f.discrete joint distribution:continuous joint distribution:Example: Marginal p.f.The joint p.f. of education level and mont

15、hly personal e is specified as: Y 123410.20.10.060.04X 20.090.06 0.10.1530.010.030.080.08Example: Marginal p.d.f.Suppose the joint p.d.f. of X and Y is: What are and ?Solution: The joint p.d.f. of X and Y is:For x1,Otherwise, For y1,Otherwise, marginal d.f. joint d.f. ?marginal p.f. or p.d.f. joint

16、p.f. or p.d.f.?IMPOSSIBLE without further information!Independent Random VariablesTwo random variables X and Y are independent if, for any two sets A and B of real numbers, X and Y are independentIf X and Y have either a discrete joint distribution or a continuous joint distribution,X and Y are independent ExampleSuppose X and Y independently have the following p.d.f.SolutionX and Y independently have the following p.d.f.Solution:Suppose X and Y have d