Topics: Statistics - Probability - Standard Normal Distribution


Let be a sequence of iid random variables (random sample) such that :

Let be the sum of all of these random variables:

From the properties of the expected value and those of variance, it follows that and . Now, let:

This is standarisation, so and .

The central limit theorem tells us that when , the sequence converges (in probability distribution) towards a standard normal distribution .



The usefulness of the central limit theorem becomes evident when, given , we use it to affirm that:

That is, the (standardised) sum of iid random variables with any distribution approaches a standard normal distribution when .

This allows us to use a normal standard distribution to approximate such a sum when is large enough.


If we set , then we can write: