2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ CLT not useful. $\endgroup$ – Did Jan 11 '14 at 23:39
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.