# Finding the limiting distribution

I was given an exercise to do that sounded something like this:

The Arizona football team scored $45$ goals in $19$ games in the 2007/08 season. If $y_i$ denotes the number of goals scored in the $i$-th game, consider the following model:

$Y_1,\dots,Y_n \sim \operatorname{Poisson}(\lambda)$ independently with $\lambda > 0$.

What is the limiting distribution of

$$T_n = \sqrt n\cdot\frac1{n\sqrt\lambda}\cdot\sum_{i=1}^n(Y_i-\lambda)$$

as $n\to\infty$?

Use $T_n$ to construct a $95$% confidence interval for $\lambda$.

Could someone please tell me what a limiting distribution is and how to find it? After that I should be able to handle the question on my own.

Thanks!

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I tried to convert your ASCII to $\LaTeX$, but I’m not sure that I interpreted it correctly; there was at least one missing parenthesis. Would you check and make any necessary corrections? – Brian M. Scott Feb 3 '13 at 0:56