[Math] Central limit theorem(Expectation)

central limit theoremexpectationprobability

I have a problem with usage of expectation in Central limit theorem.As example, look at this problem:

A certain component is critical to operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is $100$ hours and standard deviation $30$ hours, how many of these components must be in stock so that probability that system is in continual operation for the next $2000$ hours is at least $0.95$?

As we know: $E[\frac{(X_1 + X_2 + … X_n)}{n}] = u$

BUT the solution to this problem is: $P( Z < \frac{2000-100n}{30\sqrt{n}}) = 0.05$, where $n$ is the number of components.

So the question: Why do we multiply our mean ($100$ hours) by $n$?
And in which cases do we not multiply the mean by $n$?

Thanks

Best Answer

Note that

$$ \sqrt{n} \frac {\frac {1} {n} \sum_{i=1}^n X_i - \mu} {\sigma} = \frac {\sum_{i=1}^n X_i - n\mu} {\sqrt{n}\sigma}$$

So both form are equivalent.

LHS has a better interpretation as the terms inside the fraction are finite, and the fraction converge to $0$ by LLN. But CLT tell us that by scaling with $\sqrt{n}$, the fraction does not goes to infinity; instead it has a normal distribution which is finite almost surely.

Related Question