Given the random vector xx1x2 where x1 and x2 are random...

Given the random vector X=(X1,X2) where X1 and X2 are random variables with expected value mu, variance 1, and correlation rho. a) Obtain the principal components of X from the correlation matrix. b) If rho=0.7, calculate the variability retained by each principal component.

Transcript

00:01 Hello everyone.

00:02 So in the question we have given that w is equal to 2 times of x3 minus 3 times of x4 plus 2 times of x6.

00:15 We have to calculate the value of b w is also given to 02.

00:22 So we have to calculate the value of row.

00:27 So here we will write that w is equals to 2 times of x2 minus 2 times of this is x3.

00:38 3 times of x 4 plus 2 times of x 6 so v w is equals to 4 times of v multiplied with x 3 plus 9 times of v x 4 plus 4 times of v x 6 minus writing it here there is no space so 12 c ov multiplied with x 3 x 3 x 4 minus 12 times of c ov multiplied with x 4 x6 plus 8 times of co x 3 x6.

01:29 So from here substituting the values we get v w is equals to 4 times of 20 plus 9 times of 22 plus 4 times of 12 minus 12 times of 5 minus writing it here 12 times of 4 plus 8 times 4 after calculation we found out that v w is equals to 80 plus 198 plus 48 minus 60 minus 66 plus 32 so we of double we get 385 minus 156 equals to 202 so v of w is equals to 202 now in the next part v is equal to 3 times of x1 plus x2 minus 2 times of x5 so v of v we get 9 v multiplied with x1 plus v multiplied with x2 plus 4 v multiplied with x 5 plus 6 times of c ov multiplied with x 1 x2 minus 4 times of c ov multiplied with x2 x5 minus 12 times of c ov x 1 x5 substituting the values we get v or v is equal to 9 times 15 plus 18 plus 4 times 14 plus 6 times 4 minus 4 minus 12 times 6 so after calculation we get 19 15 990 135 v times of v plus 18 plus 14 into 4 56 plus 15 into 4 56 plus 6 into 424 minus 4 into 4 16 minus 12 times 6 72 so after calculation we got 233 minus 88 which is equals to 145 in the next part c o v w is equals to c o v w is equals to c o v w is equal to c o v 2 x3 minus 3 x4 plus 2 x6 and 3 x1 writing it here plus x2 minus 2 x5 so c ov v w equals to 6 c o v x 3 x1 plus 2 times of cov x3 x2 minus 4 cov x3 x5 minus 9 c ov x4 x1 minus there is a shortage of space so we need press space c3 cov x4 x4 x1 minus c3 cov x4 x4 2 plus 6 c .o .v.

06:02 X4 x5 plus 6 c .o.

06:09 V.

06:10 X6 x1 plus 2 c .o.

06:17 X6 x2 minus 4 c .o .v...

Question

Given the random vector X=(X1,X2) where X1 and X2 are random variables with expected value mu, variance 1, and correlation rho. a) Obtain the principal components of X from the correlation matrix. b) If rho=0.7, calculate the variability retained by each principal component.

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