Central Limit Theorem
For i.i.d. with mean and variance , the standardised sums converge in distribution to a standard normal:
Proof via moment generating functions
Standardise first: has and , and , so it is enough to treat i.i.d. summands of mean and variance .
Assume for some and write . By the continuity property for MGFs it suffices to show
Independence factorises the transform, , and
The tail satisfies as : for it is bounded using by a constant multiple of . Hence
which is convergence in distribution to .
Normal approximation for sums
Let be i.i.d. random variables with mean and variance . Set . Then
Examples
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Suppose . Then where are i.i.d. with . So
Therefore, for large ,
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Suppose with . Then
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We can approximate Poisson distribution with normal distribution. Suppose with . Then where are i.i.d. with . So
Therefore, for large ,
Related
Stated in
- Theorem 5.8 (Central Limit Theorem)§5.2 Central Limit Theorem
- Corollary 5.10§5.2 Central Limit Theorem
- Example 5.11§5.2 Central Limit Theorem