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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class11  >>  Statistics
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$x$ and $y$ are two correlated variables with the same standard deviation and the correlation coefficient $r$ , the correlation coefficient between $x$ and $x+y$ is

$\begin {array} {1 1} (A)\;\large\frac{r}{2} & \quad (B)\;\sqrt{\large\frac{1-r}{2}} \\ (C)\;\sqrt{\large\frac{1+r}{2}} & \quad (D)\;0 \end {array}$

 

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1 Answer

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Let $u=x+y$
$ \sigma = $ SD of $ x\: or \: y$
$ \therefore \sigma_u^2= \sigma_x^2+\sigma_y^2+2\: cov\: (x,y)$
$ = \sigma^2+\sigma^2 2r+ \sigma^2$
$ = 2\sigma^2(1+r)$
cov $(u,r) = E \{ ( u - \overline u)\: (x-\overline x) \}$
$ = E \{ [ (x-\overline x)+(y-\overline y)] (x-\overline x) \}$
$ = E \{ (x-\overline x)^2+(x-\overline x)(y-\overline y) \}$
$ = \sigma^2(1+r)$
Thus $r(x, x+y)=\large\frac{\sigma^2(1+r)}{\sigma.\sigma \sqrt{2(1+r)}}$
$ = \large\frac{\sqrt{r+1}}{2}$
Hence Ans : (C)
answered Jan 31, 2014 by thanvigandhi_1
 

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