Question

Let X, X₂ X3 X4 ~ Np (M, E) random vectors. a) Find the marginal distribution for each of the random rectors V = 3x + x +4x,-4 xe V2 = 13 x + 1 3 X 2 - 1 3 X3 - 1 344 b) Find the distribution of the random vectors V, and V2. c) Find the joint density of the random vectors V, and a defined in ca) and (b).

Answer 1

Safont civen Randem vectuos Vi= £X , - / X 2 + 1 2 13* tatu V2 = 1 3 2 - 1 2 1 2 7 3 23 - 314 @ Jirst calculate meon & Covasionce of the Vector . The meam & Covasionce of vector u, is VINNO [ 26; 0; (3 (2) {] Where 63 = (*--*-4cm) { coefficient of Veebox V) Nom meam & covosince of the Vandern vectory M(v.) = , C3 U5 = ( 22 Co 43 = (1 - 114 hlu zo Covariance is given as a la rith # +) € - LUE Here VNNO, IE) and V, is marginally distributed with mean m=0 & Co-vasionce it M banned with a Scanner

Now mean& variance for Va is Vz wone big ba az. (E bi%) ) these bj = { brze - men of rondom variable Vis a Hilva) - v bist; - +-$-}) a Coverione of vector vzis cov Witátás +) Honce verimli u, ) vz is marginally distrikter with mean M= 4/3 & covariance is of ② now the joint distribution of the Nondon vector i, ah (.] «N:8 (C ). s) where E= [ (6) € breen I bre (2) e Scanned with CamScanner

- [ 4 ] 3. Ve=C+*+( 37 bica The Joint distribution of randon vectors V, ale is C.] mw5{ [0). [4] Hence vector 1.180.2 are independent ics Scanned with CamScanner

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