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Unformatted text preview: Lecture 15 15.1 Orthogonal transformation of standard nor- mal sample. Consider X 1 , . . . , X n ∼ N (0 , 1) i.d.d. standard normal r.v. and let V be an orthogonal transformation in n . Consider a vector ~ Y = ~ XV = ( Y 1 , . . . , Y n ). What is the joint distribution of Y 1 , . . . , Y n ? It is very easy to see that each Y i has standard normal distribution and that they are uncorrelated. Let us check this. First of all, each Y i = n X k =1 v ki X k is a sum of independent normal r.v. and, therefore, Y i has normal distribution with mean 0 and variance Var( Y i ) = n X k =1 v 2 ik = 1 , since the matrix V is orthogonal and the length of each column vector is 1 . So, each r.v. Y i ∼ N (0 , 1) . Any two r.v. Y i and Y j in this sequence are uncorrelated since Y i Y j = n X k =1 v ik v jk = ~v i ~v j = 0 since the columns ~v i ⊥ ~v j are orthogonal. Does uncorrelated mean independent? In general no, but for normal it is true which means that we want to show that Y ’s are i.i.d. standard normal, i.e....
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.
- Spring '09