RS – 4 – Multivariate Distributions Chapter 4 Multivariate distributions k ≥ 2 Multivariate Distributions All 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 discrete F(x1, x2, …, xk ) when the RVs are continuous 1 RS – 4 – Multivariate Distributions Joint Probability Function Definition: Joint Probability Function Let X1, X2, …, Xk denote k discrete random variables, then p(x1, x2, …, xk ) is joint probability function of X1, X2, …, Xk if 1. 0px1 , , xn 1 2. px 1, , xn 1 xx1 n 3. P XXA11,,nn pxx ,, x1,, xAn Joint Density Function Definition: Joint density function Let X1, X2, …, Xk denote k continuous random variables, then n f(x1, x2, …, xk) = δ /δx1,δx2, …,δxk F(x1, x2, …, xk) is the joint density function of X1, X2, …, Xk if 1. fx1, , xn 0 2. fx,, xdx ,, dx 1 11nn A 3. P XXA,, fxxdxdx ,, ,, 111nnn 2 RS – 4 – Multivariate Distributions Example: The Multinomial distribution Suppose that we observe an experiment that has k possible outcomes {O1, O2, …, Ok } independently n times. Let p1, p2, …, pk denote probabilities of O1, O2, …, Ok respectively. Let Xi denote the number of times that outcome Oi occurs in the n repetitions of the experiment. Then the joint probability function of the random variables X1, X2, …, Xk is n! xx12 xk px112,, xnk pp p xx12!! xk ! Example: The Multinomial distribution xx12 xk Note: p12pp k is the probability of a sequence of length n containing x1 outcomes O1 x2 outcomes O2 … xk outcomes Ok n! n xx12!! xk !x12xx k is the number of ways of choosing the positions for the x1 outcomes O1, x2 outcomes O2, …, xk outcomes Ok 3 RS – 4 – Multivariate Distributions Example: The Multinomial distribution nnx 1 nx12 x xk xx12 x3 xk n! nx!! nx x 112 xnx112123123!!! x nx x !! x nx x x ! n! x12!!xx k ! n! xx12 xk px112,, xnk pp p xx12!! xk ! n xx12 xk pp12 pk xx12 xk This is called the Multinomial distribution Example: The Multinomial distribution Suppose 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 = 0.30, p2 = 0.50, p3 = 0.20. 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 reaction. Find the distribution of X, Y and Z. Compute P[X + Y ≥ Z] 4! xyz pxyz, , 0.30 0.50 0.20 x y z 4 xyz!!! 4 RS – 4 – Multivariate Distributions z Table: p(x,y,z) x y 01234 0 0 0 0 0 0 0.0016 0 1 0 0 0 0.0160 0 0 2 0 0 0.0600 0 0 0 3 0 0.1000 0 0 0 0 4 0.0625 0 0 0 0 1 0 0 0 0 0.0096 0 1 1 0 0 0.0720 0 0 1 2 0 0.1800 0 0 0 1 3 0.1500 0 0 0 0 1400000 2 0 0 0 0.0216 0 0 2 1 0 0.1080 0 0 0 2 2 0.1350 0 0 0 0 2300000 2400000 3 0 0 0.0216 0 0 0 3 1 0.0540 0 0 0 0 3200000 3300000 3400000 4 0 0.0081 0 0 0 0 4100000 4200000 4300000 4400000 z P [X + Y ≥ Z] x y 01234 0 0 0 0 0 0 0.0016 0 1 0 0 0 0.0160 0 = 0.9728 0 2 0 0 0.0600 0 0 0 3 0 0.1000 0 0 0 0 4 0.0625 0 0 0 0 1 0 0 0 0 0.0096 0 1 1 0 0 0.0720 0 0 1 2 0 0.1800 0 0 0 1 3 0.1500 0 0 0 0 1400000 2 0 0 0 0.0216 0 0 2 1 0 0.1080 0 0 0 2 2 0.1350 0 0 0 0 2300000 2400000 3 0 0 0.0216 0 0 0 3 1 0.0540 0 0 0 0 3200000 3300000 3400000 4 0 0.0081 0 0 0 0 4100000 4200000 4300000 4400000 5 RS – 4 – Multivariate Distributions Example: The Multivariate Normal distribution Recall the univariate normal distribution 2 1 x 1 2 fx e 2 the bivariate normal distribution 2 2 1 xxxxxxyy 2 1 212 xxyy fxy, e 2 21xy Example: The Multivariate Normal distribution The k-variate Normal distribution is given by: 1 1 1 2 xμ x μ fx1,, xk fx k /2 1/2 e 2 where x1 1 11 12 1k x x 2 μ 2 12 22 2k xk k 12kk kk 6 RS – 4 – Multivariate Distributions Marginal joint probability function Definition: Marginal joint probability function Let X1, X2, …, Xq, Xq+1 …, Xk denote k discrete random variables with joint probability function p(x1, x2, …, xq, xq+1 …, xk ) then the marginal joint probability function of X1, X2, …, Xq is pxx,, pxx,, 12qq 1 1 n xxqn1 When X1, X2, …, Xq, Xq+1 …, Xk is continuous, then the marginal joint density function of X1, X2, …, Xq is f xx,, fxxdxdx,, 12qq 1 1 nqn 1 Conditional joint probability function Definition: Conditional joint probability function Let X1, X2, …, Xq, Xq+1 …, Xk denote k discrete random variables with joint probability function p(x1, x2, …, xq, xq+1 …, xk ) then the conditional joint probability function of X1, X2, …, Xq given Xq+1 = xq+1 , …, Xk = xk is px 1 ,, xk pxxxx11qq k 11,,qq ,, k p xx,, qkq11 k For the continuous case, we have: fx ,, x fxxxx,, ,, 1 k 11qq k 11qq k f xx,, qkq11 k 7 RS – 4 – Multivariate Distributions Conditional joint probability function Definition: Independence of sects of vectors Let X1, X2, …, Xq, Xq+1 …, Xk denote k continuous random variables with joint probability density function f(x1, x2, …, xq, xq+1 …, xk ) then the variables X1, X2, …, Xq are independent of Xq+1, …, Xk if f xxfxxfx,, ,, ,, x 11111 kq qqkqk A similar definition for discrete random variables. Conditional joint probability function Definition: 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 independent if f xxfxfxfx11122,, kkk A similar definition for discrete random variables. 8 RS – 4 – Multivariate Distributions Multivariate marginal pdfs - Example Let X, Y, Z denote 3 jointly distributed random variable with joint density function then 2 Kx yz01,01,01 x y z fxyz,, 0otherwise Find the value of K. Determine the marginal distributions of X, Y and Z. Determine the joint marginal distributions of X, Y X, Z Y, Z Multivariate marginal pdfs - Example Solution: Determining the value of K. 111 1,, f x y z dxdydz K x2 yz dxdydz 000 x1 11x3 111 K xyz dydz K yz dydz 0033x0 00 y1 11111y 2 KyzdzKzdz 0032 32 12 y0 if K 1 7 zz2 11 7 KKK 1 340 34 12 9 RS – 4 – Multivariate Distributions Multivariate marginal pdfs - Example The marginal distribution of X. 12 11 f x f x,, y z dydz x2 yz dydz 1 7 00 112 y 1 1222y 12 1 x y z dz x z dz 7200y 0 72 2 1 1222z 12 1 x zx for 0 x 1 74740 Multivariate marginal pdfs - Example The marginal distribution of X,Y. 12 1 f xy,,, f xyzdz x2 yzdz 12 7 0 2 z1 12 2 z xz y 72z0 122 1 x yxy for 0 1,0 1 72 10 RS – 4 – Multivariate Distributions Multivariate marginal pdfs - Example Find the conditional distribution of: 1. Z given X = x, Y = y, 2. Y given X = x, Z = z, 3. X given Y = y, Z = z, 4. Y , Z given X = x, 5. X , Z given Y = y 6. X , Y given Z = z 7. Y given X = x, 8. X given Y = y 9. X given Z = z 10. Z given X = x, 11. Z given Y = y 12. Y given Z = z Multivariate marginal pdfs - Example The marginal distribution of X,Y. 122 1 f12 xy, x y for 01,01 x y 72 Thus the conditional distribution of Z given X = x,Y = y is 12 2 fxyz,, x yz 7 fxy, 122 1 12 x y 72 xyz2 for 0z 1 1 xy2 2 11 RS – 4 – Multivariate Distributions Multivariate marginal pdfs - Example The marginal distribution of X. 122 1 f1 xx for 0 x 1 74 Then, the conditional distribution of Y , Z given X = x is 12 2 fxyz,, x yz 7 fx 122 1 1 x 74 xyz2 for 0yz 1,0 1 1 x2 4 Expectations for Multivariate Distributions Definition: Expectation Let X1, X2, …, Xn denote n jointly distributed random variable with joint density function f(x1, x2, …, xn ) then EgX1 ,, Xn g x,, x f x ,, x dx ,, dx 111nnn 12 RS – 4 – Multivariate Distributions Expectations for Multivariate Distributions - Example Let X, Y, Z denote 3 jointly distributed random variable with joint density function then 12 2 7 x yz01,01,01 x y z fxyz,, 0otherwise Determine E[XYZ]. Solution: 111 12 E XYZ xyz x2 yz dxdydz 000 7 12 111 x 322yz xy z dxdydz 7 000 Expectations for Multivariate Distributions - Example 111 12 12 111 E XYZ xyzx2 yzdxdydz x 322yz xy z dxdydz 7 000 7 000 1142x 1 11 12xx22 3 22 yz y zdydz yz 2 y zdydz 7400 2x 0 7 00 1123y 1 3312yy22 zzdzzzdz2 7200 3y 0 723 1 32zz23 31231717 74 90 749 73684 13 RS – 4 – Multivariate Distributions Some Rules for Expectations – Rule 1 1.