Suppose X 1 , … , X m and Y 1 , … , Y n are random variables with the property that C o v ( X i , Y j ) &lt; ∞ for all i and j. Show that, for any constants a 1 , … , a m and b 1 , … , b n , C o v ( ∑ i = 1 m a i X i , ∑ j = 1 n b j Y j ) = ∑ i = 1 m ∑ j = 1 n a i b j C o v ( X i , Y j )

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Suppose       X    1    ,  …  ,      X    m   and       Y    1    ,  …  ,      Y    n   are random variables with the property that    C  o  v  (      X    i    ,      Y    j    )  &amp;lt;  ∞ for all i and j. Show that, for any constants       a    1    ,  …  ,      a    m   and       b    1    ,  …  ,      b    n  ,  C  o  v  (      ∑          i      =      1        m        a    i        X    i    ,      ∑          j      =      1        n        b    j        Y    j    )  =      ∑          i      =      1        m        ∑          j      =      1        n        a    i        b    j    C  o  v  (      X    i    ,      Y    j    )

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→      x    1    ,  …  ,      x    m   and       y    1    ,  …  ,      y    n   are random variable with the property that _  c  o  v  (      x    i    ,      y    j    )  &amp;lt;  ∞for all i and j. Show that _for any constants        a    1    ,  …  ,      a    m     and       b    1    ,  …      b    n    c  o  v  (      ∑          i      =      1        m        a    i        x    i    ,      ∑          j      =      1        n        b    j        Y    j    )  =      ∑          i      =      1        m        ∑          j      =      1        n        a    i        b    j    c  o  v  (      x    i    ,      Y    j    )  c  o  v  (      ∑          i      =      1        m        a    i        x    i    ,      ∑          j      =      1        n        b    j        Y    j    )  =      ∑          i      =      1        m        ∑          j      =      1        n        a    i        b    j    (  c  o  v  (      a    i        x    i    ,      b    j        Y    j    )  )  =      ∑          i      =      1        m        ∑          j      =      1        n        a    i        b    j    c  o  v  (      x    i    ,      Y    j    )Hence, show that for any constant       a    1    ,  …  ,      a    m     and       b    1    ,  …  ,      b    n    c  o  v  (      ∑          i      =      1        m        a    i        x    i    ,      ∑          j      =      1        m        b    j        Y    j    )  =      ∑          i      =      1        m        ∑          j      =      1        n        a    i        b    j    c  o  v  (      X    i    ,      Y    j    )

Suppose X_1,...,X_m and Y_1,...,Y_n are random variables with the property that cov(X_i,Y_i)<oo for all i and j. Show that, for any constant a_1,..., a_m and b_1,...,b_n, Cov(sum_(i=1)^m a_iX_i,sum_(j=1)^n b_jY_j)=sum_(i=1)^m sum_(j=1)^n a_i b_j Cov(X_i,Y_j)

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