5.6 The Central Limit Theorem

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Presentation transcript:

5.6 The Central Limit Theorem There are many cases for which we know the cdf and density of the sum (i.i.d.) For example the Bernoulli, binomial, Poisson, gamma, and Gaussian.

Preliminary Observations If m ≠ 0, nm → + or – ∞.

So, we might consider instead

Xi ∼ exp(1) RVs implies Yn ∼ Erlang(n,1)

Derivation of the CLT

Chapter 7 Bivariate RVs

Marginal Probabilities

Marginal Probabilities

Marginal Probabilities

Marginal Probabilities

Marginal Probabilities Similarly,

7.2 Jointly Continuous RVs

Joint and Marginal Densities

Conversely, if