caffeine

## Main Question or Discussion Point

I don't think I *really* understand the Central Limit Theorem.

Suppose we have a set of [itex]n[/itex] independent random variables [itex]\{X_i\}[/itex] with the same distribution function, same finite mean, and same finite variance. Suppose we form the sum [itex]S_n = \sum_{i=1}^n X_i[/itex]. Suppose I want to know the probability that [itex]S_n[/itex] is between a and b. In other words, I want to know [itex]P(b > S_n > a)[/itex].

The central limit theorem uses a

[tex]

P\left(b > \frac{ \sum_{i=1}^n X_i - n\mu}{\sqrt{n}\sigma} > a\right)

= \frac{1}{\sqrt{2\pi}} \int_a^b e^{-y^2/2} \, dy

[/tex]

What is the relationship between what I want:

[tex]P(b > S_n > a)[/tex]

and what the central limit theorem tells me about:

[tex]P\left(b > \frac{ \sum_{i=1}^n X_i - n\mu}{\sqrt{n}\sigma} > a\right)[/tex]

How can they possibly be equal? If they are equal, how is that possible? And if they're not equal, how would I get what I want?

Suppose we have a set of [itex]n[/itex] independent random variables [itex]\{X_i\}[/itex] with the same distribution function, same finite mean, and same finite variance. Suppose we form the sum [itex]S_n = \sum_{i=1}^n X_i[/itex]. Suppose I want to know the probability that [itex]S_n[/itex] is between a and b. In other words, I want to know [itex]P(b > S_n > a)[/itex].

The central limit theorem uses a

*standardized*sum:[tex]

P\left(b > \frac{ \sum_{i=1}^n X_i - n\mu}{\sqrt{n}\sigma} > a\right)

= \frac{1}{\sqrt{2\pi}} \int_a^b e^{-y^2/2} \, dy

[/tex]

What is the relationship between what I want:

[tex]P(b > S_n > a)[/tex]

and what the central limit theorem tells me about:

[tex]P\left(b > \frac{ \sum_{i=1}^n X_i - n\mu}{\sqrt{n}\sigma} > a\right)[/tex]

How can they possibly be equal? If they are equal, how is that possible? And if they're not equal, how would I get what I want?

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