glamrockqueen7

2021-03-18

For large value of n, and moderate values of the probabiliy of success p (roughly, $0.05\text{}\Leftarrow \text{}p\text{}\Leftarrow \text{}0.95$ ), the binomial distribution can be approximated as a normal distribution with expectation mu = np and standard deviation

$\sigma =\sqrt{np(1\text{}-\text{}p)}$ . Explain this approximation making use the Central Limit Theorem.

Asma Vang

Skilled2021-03-19Added 93 answers

Step 1

Alternatively prove that:

Theorem:

If x is a random variable with distribution B(n, p), then for sufficiently large n, the distribution of the

variablez

where

Proof:

It can be prove using Moment generating function for binomial distribution. It's given as,

where

Step 2

By the linear transformation properties of the moment generating function.

Taking the natural log of both sides, and then expanding the power series of

Then,

Since

If n is made sufficiently large

Let

Thus for sufficiently large

The ln term in the previous expression is

Step 3

This means that

42

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