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48is σ 2 Y .Independence CaseWhen β = 0, the Y i are independent. In this case the following statements aretrue:E[Y i ] = C (4.2)V AR[Y i ] = σ 2 Y = σ2 ɛ (4.3)E[Ȳ ] = E[Y i ] = C (4.4)V AR[Ȳ ] =(σ2YN)=( )σ2ɛN(4.5)Furthermore, we can write down the loglikelihood by using the usual productof independent Gaussian densities.L(Y |M) = − N 2 log(2π) − N 2 log(σ2 Y ) − 1 N∑(Y2σY2 i − C) 2 (4.6)i=1Because we will later want to condition on the value of Y 1 , we do this here also:L(Y −1 |M, Y 1 ) = − N − 12log(2π) − N − 1 log(σY 2 ) − 122σY2N∑(Y i − C) 2 (4.7)i=2As N increases, the contribution of Y 1 to the loglikelihood becomes proportionatelysmaller.Equations 4.2 to 4.5 are actually special cases of equations 4.8 to 4.11, whichare shown in the next section. When β = 0 is substituted into equations 4.8 to4.11, they reduce to equations 4.2 to 4.5.