De Moivre–Laplace theorem
The set of Moivre -Laplace is a set of probability theory. After this block, the binomial probabilities for and converges to the normal distribution. For large sample size can therefore be used as an approximation to the binomial distribution, the normal distribution, which in particular applies in hypothesis testing.
In the set of Moivre - Laplace, who is named after Abraham de Moivre and Pierre -Simon Laplace, it is a special case of the central limit theorem.
Statement
Is a binomial random variable with parameters and then applies
By means of a substitution can be seen that
Stands for the probability distribution function of the standard normal distribution, which is also referred to as Gaussian error integral. Values for one usually takes a table.
The set of Moivre -Laplace provides sufficiently good approximations if they meet the following condition:
Example
Given a binomial distribution with and so therefore applies. We compare it to a normal distribution with mean and variance.
Applications
The set of Moivre -Laplace is the theoretical basis of normal approximation, a method by which the binomial distribution can be approximated