Suppose that $X_1,\ldots,X_n$ is an iid sample from the Poisson distribution with mean $\lambda$. Use the Central Limit theorem to find $P(|\bar X - \lambda| < 0.1) $ as $n$ goes to infinity.

My question is, if $n$ goes to infinity, the variance for $\bar X$ would be zero and it does not make any sense to me.

  • $\begingroup$ CLT not useful. $\endgroup$ – Did Jan 11 '14 at 23:39

For finite n, we have

$var(\bar X) > 0 $,

but $var(\bar X) \rightarrow 0$ for $n \rightarrow \infty$

Since $\bar X \rightarrow \lambda\ ,$

the probability you mentioned tends to 1, because for some n,

we have $P(|\bar X-\lambda|<0.1) > 1-\epsilon$

for each given $\epsilon>0$

Or, shortly formulated :

$$\lim_{n \rightarrow \infty} P(|\bar X-\lambda|<0.1)=1$$

This type of convergence is called convergence in probability.

For large $n$, $\bar X$ is nearly normal-distributed and since the variance tends to 0, $\bar X$ tends to the constant random-variable having the value $\lambda$ with probability $1$.

The $0.1$ can, by the way, be replaced by any positive value.


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