Chapter 4 Multivariate distributions - Bauer College of ...

1 RS 4 Multivariate Distributions1 Chapter 4 Multivariate distributionsk 2 Multivariate DistributionsAll the results derived for the bivariate case can be generalized to n RV. The joint CDF of X1, X2, .., Xk will have the form: P(x1, x2, .., xk) when the RVs are discreteF(x1, x2, .., xk) when the RVs are continuousRS 4 Multivariate Distributions2 Joint Probability FunctionDefinition: Joint Probability FunctionLet X1, X2, .., Xk denote k discrete random variables, then p(x1, x2, .., xk) is joint probability function of X1, X2, .., Xk if 112. ,,1nnxxpxx 11. 0,,1npxx 113. ,,,,nnPXXApxx 1,,nxxA Definition: Joint density function Let X1, X2, .., Xk denote k continuous random variables, then f(x1, x2, .., xk) = n/ x1, x2, .., xkF(x1, x2, .., xk)is the joint density function of X1, X2.

2 , Xk if 112. ,,,,1nnfxx dxdx 11. ,,0nfxx 1113. ,,,,,,nnnPXXA fxxdxdx AJoint Density FunctionRS 4 Multivariate Distributions3 Example: The Multinomial distributionSuppose that we observe an experiment that has k possible outcomes {O1, O2, .., Ok} independently n p1, p2, .., pk denote probabilities of O1, O2, .., Okrespectively. Let Xidenote the number of times that outcome Oioccurs in the n repetitions of the the joint probability function of the random variables X1, X2, .., Xkis 1211212! ,,!! !kxxxnkknpxxp ppxx x Note:is the probability of a sequence of length ncontainingx1outcomes O1x2outcomes outcomes Ok1212kxxxkppp 1212!!! !kknnxxxxx x is the number of ways of choosing the positions for the x1outcomes O1, x2outcomes O2.

3 , xk outcomes OkExample: The Multinomial distributionRS 4 Multivariate Distributions4121312kknx xxnnxxxxx 112112 123123!!!!!!!!!nxnx xnx nxx nx xx nx x x 12!!!!knxxx This is called the Multinomialdistribution 1211212! ,,!! !kxxxnkknpxxp ppxx x 121212kxxxkknpppxxx Example: The Multinomial distributionSuppose that an earnings announcements has three possible outcomes:O1 Positive stock price reaction (30% chance)O2 No stock price reaction (50% chance)O3- Negative stock price reaction (20% chance)Hence p1= , p2= , p3= Suppose today 4 firms released earnings announcements (n = 4). Let X = the number that result in a positive stock price reaction, Y= the number that result in no reaction, and Z= the number that result in a negative the distribution of X, Y and Z.

4 Compute P[X +Y Z] 4! , , 4!!!xyzpxyzx y zxyz Example: The Multinomial distributionRS 4 Multivariate Distributions5 Table: p(x,y,z) [X + Y Z] 4 Multivariate Distributions6 Example: The Multivariate Normal distributionRecall the univariate normal distribution 2121 2xfxe the bivariate normal distribution 221221221 ,21xxxxyyxxxxyyxyfxye Thek-variate Normal distributionis given by: 1121/21/ 21 ,,2kkfxxfe x x x where12 kxxx x 12 k 111211222212 kkkkkk Example: The Multivariate Normal distributionRS 4 Multivariate Distributions7 Marginal joint probability functionDefinition: Marginal joint probability function Let X1, X2, .., Xq, Xq+ , Xk denote k discrete random variables with joint probability function p(x1, x2.)

5 , xq, xq+ , xk)then the marginal joint probability functionof X1, X2, .., Xq is 11211 ,, ,,qnqqnxxpxxpxx When X1, X2, .., Xq, Xq+ , Xk is continuous, then the marginal joint density functionof X1, X2, .., Xq is 12111 ,, ,,qqnqnfxxfxxdx dx Conditional joint probability functionDefinition: Conditional joint probability functionLet X1, X2, .., Xq, Xq+ , Xk denote k discrete random variables with joint probability function p(x1, x2, .., xq, xq+ , xk)then the conditional joint probability functionof X1, X2, .., Xq given Xq+1 = xq+1 , .., Xk= xkis 1111111,, ,,,, ,,kqqkqqkqkqkpxxpxxxxpxx 1111111,, ,,,, ,,kqqkqqkqkqkfxxfxxxxfxx For the continuous case, we have: RS 4 Multivariate Distributions8 Definition: Independence of sects of vectors Let X1, X2.

6 , Xq, Xq+ , Xk denote k continuous random variables with joint probability density function f(x1, x2, .., xq, xq+ , xk) then the variables X1, X2, .., Xq are independentof Xq+1, .., Xkif 11111 ,, ,, ,,kqqqkqkfxxfxxf x x A similar definition for discrete random joint probability functionDefinition: Mutual Independence Let X1, X2, .., Xk denote k continuous random variables with joint probability density function f(x1, x2, .., xk) then the variables X1, X2, .., Xk are called mutually independentif 11122 ,, kkkfxxfxfx fx A similar definition for discrete random joint probability functionRS 4 Multivariate Distributions9 Multivariate marginal pdfs - ExampleLet X, Y, Z denote 3 jointly distributed random variable with joint density function then 201,01,01,,0otherwiseKx yzxyzfxyz Find the value of the marginal distributions of X, Y and the joint marginal distributions of X, YX, ZY, Z 11120001,,fx y z dxdydzK xyz dxdydz Solution.

7 Determining the value of dydz 11120001113232yyyKyzdzK zdz 1201171343412zzKKK 12if 7K Multivariate marginal pdfs - ExampleRS 4 Multivariate Distributions10 11210012,,7fxf x y z dydzxyz dydz The marginal distribution of 72yyyxyzdzxz dz 1222012121 for 017474zxzx x Multivariate marginal pdfs - Example 1212012,,,7fxyf xyzdzx yzdz The marginal distribution of X, y 2121 for 01, 0172xyxy Multivariate marginal pdfs - ExampleRS 4 Multivariate Distributions11 Find the conditional distribution given X = x, Y = y, given X = x, Z = z, given Y = y, Z = z, , Z given X = x, , Z given Y = y , Y given Z = given X = x, given Y = y given Z = given X = x, given Y = y given Z = zMultivariate marginal pdfs - ExampleThe marginal distribution of X,Y.

8 212121, for 01,0172fxyxyxy Thus the conditional distribution of Z given X = x,Y = y is 221212,,7121,72xyzfxyzfxyxy 22 for 0112xyzzxy Multivariate marginal pdfs - ExampleRS 4 Multivariate Distributions12 The marginal distribution of X. 21121 for 0174fxxx Then, the conditional distribution of Y , Z given X = x is 22112,,712174xyzfxyzfxx 22 for 01, 0114xyzyzx Multivariate marginal pdfs - ExampleExpectations for Multivariate DistributionsDefinition: ExpectationLet X1, X2, .., Xn denote n jointly distributed random variable with joint density function f(x1, x2, .., xn) then 1,,nEgXX 111,,,,,,nnng xxf xxdxdx RS 4 Multivariate Distributions13 Let X, Y, Z denote 3 jointly distributed random variable with joint density function then 212701,0 1,01,,0otherwisexyzxyzfxyz Determine E[XYZ].

9 Solution:Expectations for Multivariate distributions -Example 1112000127E XYZxyzxyz dxdydz 111322000127xyzxy zdxdydz 1112000127 EXYZ xyzxyzdxdydz 1111142222200000123274 27xxxxyzyzdydzyzyzdydz 11123220003312272372 3yyyyzzdz zzdz 123032 3123171774 974 9 736 84zz Expectations for Multivariate distributions -Example 111322000127xyzxy zdxdydz RS 4 Multivariate Distributions14 111. ,,iinnEXxfxxdxdx iiiixfxdx Thus you can calculate E[Xi] either from the joint distribution of X1, .. , Xn or the marginal distribution of : 11,,,,in nxfxx dxdx 1111,,iniinixfxx dx dxdxdx dx ii iixfxdx Some Rules for Expectations Rule 1 11112. nnnnEaXaXaE XaE X This property is called the Linearity property. Proof: 1111,,nnnnaxa x f xx dxdx 1111,,nnaxfxxdxdx 11,,nnn naxfxxdxdx Some Rules for Expectations Rule 2RS 4 Multivariate Distributions15 11,,,,qqkEgXX hXX In the simple case when k = 2 , and g(X)=X& h(Y)=Y:3.

10 (The Multiplicative property)Suppose X1, .. , Xq are independent of Xq+1, .. , Xk then 11,,,,qqkEgXX EhXX EXYE X E Y if X and Y are independent Some Rules for Expectations Rule 3 11,,,,qqkEgXX hXX Proof: 1111,,,,,,qqkkngxx hxx f xx dx dx 1111,,,,,,qqkqgxx hxx f xx 2111,,qkqqkfxx dxdx dxdx 1111,,qqqkfxx dxdx dxdx 1211,,,,,,qkqkqhxx f xxgxx Some Rules for Expectations Rule 3 1211,,,,qkqkqkh xxfxx dxdx 1,,qEgXX RS 4 Multivariate Distributions16 11,,,,qqkEgXX EhXX 1211,,,,qkqkqkh xx f xx dxdx 1,,qEgXX Some Rules for Expectations Rule 3 1. VarVarVar2 Cov,XYXY XY Proof:Thus, w h ere C o v,=XYXY E XY 2 VarXYXY E XY w here XYXYEX Y 2 VarXYXY E XY 222 XXYYEXXYY Var2 Cov,VarXXYY Some Rules for Variance Rule 1RS 4 Multivariate Distributions17 and VarVarVarXYXY Note: If X and Y are independent, then Cov,=XYXY E XY =XYEXEY =0 XYEXEY Some Rules for Variance Rule 1 22and V ar2 XYXYXYXY Definition: Correlation coefficient For any two random variables X and Y then define thecorrelation coefficient XY to be: Cov,Cov,= VarVarxyXYXYXYXY Thus Cov,= XYXYXY if X and Y are Some Rules for Variance Rule 1 - XYRS 4 Multivariate Distributions18 Recall Cov,Cov,= VarVarxyXYXYXYXY Property 1.