Let X_{1}....,X_{n} and Y_{1},...,Y_{m} be two sets of random variables. L

EunoR

EunoR

Answered question

2021-11-06

Let X1.,XnandY1,,Ym be two sets of random variables. Let ai,bj be arbitrary constant.
Show that
Cov(i=1naiXi,j=1mbjYj)=i=1nj=1maibjCov(Xi,Yj)

Answer & Explanation

Mitchel Aguirre

Mitchel Aguirre

Skilled2021-11-07Added 94 answers

Step 1:
It is given that X1,,XnandY1,,Ym be the two sets of random variables.
Let ai,bj be the arbitrary constant.
It is ed to show,
Cov(i=1naiXi,j=1mbjYj)=i=1nj=1maibjCov(Xi,Yj)
Step 2
Proof :
Cov(i=1naiXi,j=1mbjYj)
=E(i=1naiXi,j=1mbjYj)E(i=1naiXi)E(j=1mbjYj)
=Ei=1nj=1maibj[E(XiYj)=E(Xi)E(Yj)]
=Ei=1nj=1maibj C(Xi,Yj).bycovarianceertyCov(X,Y)=E(XY)E(X)E(Y).
Hence , It is proved.

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