Let

(x_(1),Y_(1)),(x_(2),Y_(2)),dots,(x_(n),Y_(n))

be a random sample from a bivariate normal distribution where\

x_(i)

and

Y_(i)

are independent such that\

x_(i)N(\\\\mu _(i),\\\\phi ) and Y_(i)N(\\\\mu _(i),\\\\phi )

\ Note that each pair

(x_(i),Y_(i))

has a different mean,

\\\\mu _(i)

, but all pairs share a common variance,

\\\\phi

.\ a) Find the MLEs of the parameters,

(\\\\theta ,\\\\phi )

where

\\\\theta =(\\\\mu _(1),dots,\\\\mu _(n)),-\\\\infty

, and\

\\\\phi >0

. (Express these in terms of

x_(i)

and

Y_(i)

.)\ b) Find the expected values of the

hat(\\\\mu )_(i)

. Compare this to

\\\\theta

.\ c) Find the expected value of

hat(\\\\phi )

. Why is this not equal to

\\\\phi

?

Let (X1,Y1),(X2,Y2),,(Xn,Yn) be a random sample from a bivariate normal distribution where Xi and Yi are independent such that XiN(i,)andYiN(i,) Note that each pair (Xi,Yi) has a different mean, i, but all pairs share a common variance, . a) Find the MLEs of the parameters, (,) where =(1,,n),0. (Express these in terms of Xi and Yi.) b) Find the expected values of the ^i. Compare this to . c) Find the expected value of ^. Why is this not equal to