Let X and Y be random variables having mean 0, variance 1, and correlation ρ . Show that X − ρ Y and Y are uncorrelated, and that X − ρ Y has mean 0 and variance 1 − ρ 2 .

Question

Let X and Y be random variables having mean 0, variance 1, and correlation   ρ . Show that   X  −  ρ Y and Y are uncorrelated, and that   X  −  ρ  Y has mean 0 and variance   1  −      ρ          2      .

Conner Singleton · Accepted Answer

We have that  C  o  v  (  X  −  ρ  Y  ,  Y  )  =  C  o  v  (  X  ,  Y  )  −  ρ  V  a  r  (  Y  )  =  ρ  −  ρ  =  0where we have used that   C  o  v  (  X  ,  Y  )  =  ρ  (  X  ,  Y  )  ∗      V    a    r    (    X    )    V    a    r    (    Y    )    =  ρ. So we have proved that   X  −  ρ  Y and Y are uncorrelated since the covariance is equal to zero. Further, we have that  E  (  X  −  ρ  Y  )  =  E  (  X  )  −  ρ  E  (  Y  )  =  0and that  V  a  r  (  X  −  ρ  Y  )  =  V  a  r  (  X  )  +      ρ    2    V  a  r  (  Y  )  −  2  ρ  C  o  v  (  X  ,  Y  )  =  1  +      ρ    2    −  2      ρ    2    =  1  −      ρ    2  so we have proved the claimed.

Let X and Y be random variables having mean 0, variance 1, and correlation rho . Show that X-rho Y and Y are uncorrelated, and that X-rho Y has mean 0 and variance 1-rho^(2)

Answered question

Answer & Explanation

New Questions in College Statistics