4. [25 points] Consider a sequence of Poisson random variables {X} some An > 0. (a)...

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4. [25 points] Consider a sequence of Poisson random variables {X₂} some An > 0. (a) Assume that An is a positive integer (i.e. An € {1,2,3....}). Using the central limit theorem, prove that Xn An D. √An Remark. • You may use the fact that E[X₂] = Var[X₂] variable. Dy N(0, 1) as A, → ∞0. Remark. ~ Poisson(n) for You can also use the fact that if X~ Poisson (A₁) and Y~ Poisson(₂), and they are independent, then X+Y~ Poisson (₁ + A₂). • You don't need to prove the central limit theorem. Xn - An √An = An for a Poisson random (b) Now we let An be any positive real sequence. Prove that the previous claim still remains true, i.e. D, N(0, 1) as An →∞. Slutsky's theorem would be helpful to prove this general claim. 4. [25 points] Consider a sequence of Poisson random variables {X₂} some An > 0. (a) Assume that An is a positive integer (i.e. An € {1,2,3....}). Using the central limit theorem, prove that Xn An D. √An Remark. • You may use the fact that E[X₂] = Var[X₂] variable. Dy N(0, 1) as A, → ∞0. Remark. ~ Poisson(n) for You can also use the fact that if X~ Poisson (A₁) and Y~ Poisson(₂), and they are independent, then X+Y~ Poisson (₁ + A₂). • You don't need to prove the central limit theorem. Xn - An √An = An for a Poisson random (b) Now we let An be any positive real sequence. Prove that the previous claim still remains true, i.e. D, N(0, 1) as An →∞. Slutsky's theorem would be helpful to prove this general claim.

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a Let n k where k is a positive integer Then we can write Xn Y1 Y2 Yk Where Yi Poisson... View the full answer