Chapter 4: Problem 14
Let \(\bar{X}_{n}\) denote the mean of a random sample of size \(n\) from a Poisson distribution with parameter \(\mu=1\) (a) Show that the mgf of \(Y_{n}=\sqrt{n}\left(\bar{X}_{n}-\mu\right) / \sigma=\sqrt{n}\left(\bar{X}_{n}-1\right)\) is given by \(\exp \left[-t \sqrt{n}+n\left(e^{t / \sqrt{n}}-1\right)\right]\) (b) Investigate the limiting distribution of \(Y_{n}\) as \(n \rightarrow \infty\). Hint: Replace, by its MacLaurin's series, the expression \(e^{t / \sqrt{n}}\), which is in the exponent of the mgf of \(Y_{n}\).
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.