For random variables X and Y, show: a) Cov(X + b, X) =...

For random variables X and Y, show: a) Cov(X + b, X) = Var(X) from the definition of covariance. b) Var(aX + bY) = a^2Var(X) + 2abCov(X, Y) + b^2Var(Y). Hint: You may use appropriate theorems from lectures.

Transcript

00:01 Hello students here covariance of x comma y equals to expectation of xy minus expectation of x into expectation of y this one we know so covariance of x plus b comma x so this can be written as expectation of x plus b into x minus expectation of x plus b into expectation of x so here now multiplying expectation of x square plus bx minus expectation of x into expectation of x minus b into expectation of x so here also we are multiplying so here this can be written as expectation of x square plus b into expectation of x minus expectation of x into expectation of x minus b into expectation of x so here this get cancelled so remaining expectation of x square minus expectation of x whole square so that is equal to variance of x so hence the proof and next for the next question that is b variance of ax plus by this can be written as a square into variance of x plus 2ab covariance of x comma y plus b square into variance of y this one we need to prove so first let us consider u equal to ax plus by so expectation of u is expectation of ax plus by that is a into expectation of x plus b into expectation of y so now u minus expectation of u so that is ax plus by minus a into expectation of x plus b into expectation of y so on simplifying a into or taking a common outside a into x minus expectation of x plus b into y minus expectation of y so now squaring and taking expectation on both sides squaring and taking expectation on both sides that is expectation of u minus expectation of u whole square that is equal to a square into expectation of x minus expectation of x whole square plus b square into expectation of y minus expectation of y whole square plus 2ab into expectation of x minus expectation of x into y minus expectation of y.

03:39 So using the factor that is expectation of ax plus by whole square is equal to expectation of a square x square plus b square y square plus 2ab into y that is a square into expectation of x square plus b square into expectation of y square plus 2ab expectation of xy.

04:08 So now variance of u that is a square into variance of x plus b square into variance of y plus 2ab into expectation of xy minus x into expectation of x x into expectation of y minus y into expectation of x plus expectation of x into expectation of y.

04:38 So now this is equal to a square plus variance of x plus b square into variance of y plus 2ab into expectation of xy minus expectation of x into expectation of y minus expectation of y into expectation of x plus expectation of x into expectation of y.

05:10 So here this gets cancelled.

05:14 So remaining a square plus a square into variance of x plus b square into variance of y plus 2ab into expectation of xy minus expectation of x into expectation of y...

Question

For random variables X and Y, show: a) Cov(X + b, X) = Var(X) from the definition of covariance. b) Var(aX + bY) = a^2Var(X) + 2abCov(X, Y) + b^2Var(Y). Hint: You may use appropriate theorems from lectures.

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