problem 3 a confidence intenval for poisson distributions let x1 xn be iid poisson random variab

First, we need to find two sequences $(a_n)_{n in mathbb{N}}$ and $(b_n)_{n in mathbb{N}}$ such

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Problem 3: A confidence interval for Poisson distributions Let X1, Xn be iid Poisson random variables with parameter λ > 0, and denote by Xn their empirical average: X. Find two sequences (an) and (bn) such that (an(Xn - bn)) converges in distribution to a standard Gaussian random variable Z ~ N(0,1). Prove that for all t > 0, P(|Z| < t) = 2P(Z < t) - 1. Using the previous questions, find an interval T centered around Xn such that P(T ≤ Xn) ≥ 0.95, as n approaches infinity. [Hint: The 97.5%-quantile of the standard Gaussian distribution is 1.96.] Modify the previous interval T in order to get a new interval J that is not necessarily centered around Xn, does not depend on λ, and such that P(J) ≥ 0.05, as n approaches infinity.

          Problem 3: A confidence interval for Poisson distributions

Let X1, Xn be iid Poisson random variables with parameter λ > 0, and denote by Xn their empirical average: X.

Find two sequences (an) and (bn) such that (an(Xn - bn)) converges in distribution to a standard Gaussian random variable Z ~ N(0,1). 

Prove that for all t > 0, P(|Z| < t) = 2P(Z < t) - 1.

Using the previous questions, find an interval T centered around Xn such that P(T ≤ Xn) ≥ 0.95, as n approaches infinity. [Hint: The 97.5%-quantile of the standard Gaussian distribution is 1.96.]

Modify the previous interval T in order to get a new interval J that is not necessarily centered around Xn, does not depend on λ, and such that P(J) ≥ 0.05, as n approaches infinity.

Added by Christopher A.

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