i.i.d. 4. (Pooled z-test as likelihood ratio test) Let X1,..., Xm N(1, 1) and Y₁,..., Yn Suppose that these two samples are independent. We would like to test Ho:H= H2, HH#U2 i.i.d. using the likelihood ratio test. N(μ2, 1). (a) Prove that the likelihood function can be written as 1 L(μ41,2)= exp (2π)(m+n)/2 m n {− ½Σ(ª¡ − µ1)²} exp{ − ½Σ(v; – 12)²}. i=1 j=1 (b) Prove that the maximum likelihood estimators of μ₁ and μ2 are respectively the sample mean of X and Y, that is, m Â₁ = X = X₁₁₂ == ΣX με m ΣΥ n i=1 j=1 (c) Under Ho₁₂, let us write μ₁ =μ₂ =μ. Prove that the maximum likelihood estimator of μ, assuming Ho is true, is the pooled estimator 11 ΣX + ΣY _ mX + n m+n m+n

Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.

(4) Consider n i.i.d. samples of X ~ N(µ,0²). Find the maximum likelihood estimate of o?.

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