Mathematics, 22.06.2019 04:10 toottoot42169
Random variables x poisson~ ( a) ,y poisson ~ ( a) . x and y are independent. if 2 1, 2 1. u =2x+ y-1, v=2x- y +1. find: ) cov (u ,v ).
Answers: 2
![\mathrm{Cov}(U,V)=E[(U-E[U])(V-E[V])]=E[UV-E[U]V-UE[V]+E[U]E[V]]=E[UV]-E[U]E[V]](/tpl/images/0002/7416/731b4.png)
and 
, we have![E[U]=2E[X]+E[Y]-1](/tpl/images/0002/7416/3742c.png)
![E[V]=2E[X]-E[Y]+1](/tpl/images/0002/7416/031a0.png)
![\implies E[U]E[V]=(2E[X]+E[Y]-1)(2E[X]-(E[Y]-1))=4E[X]^2-(E[Y]-1)^2=4E[X]^2-E[Y]^2+2E[Y]-1](/tpl/images/0002/7416/bb0e2.png)
![\implies E[UV]=4E[X^2]-E[Y^2]+2E[Y]-1](/tpl/images/0002/7416/06bda.png)
![\mathrm{Cov}(U,V)=(4E[X^2]-E[Y^2]+2E[Y]-1)-(4E[X]^2-E[Y]^2+2E[Y]-1)](/tpl/images/0002/7416/462bf.png)
![\mathrm{Cov}(U,V)=4(E[X^2]-E[X]^2)-(E[Y^2]-E[Y]^2)](/tpl/images/0002/7416/1127d.png)
![\mathrm{Cov}(U,V)=4V[X]-V[Y]=4a-a=\boxed{3a}](/tpl/images/0002/7416/7664e.png)
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Random variables x poisson~ ( a) ,y poisson ~ ( a) . x and y are independent. if 2 1, 2 1. u =2x+ y-...
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