Normal Approximation of the Binomial Distribution
The binomial distribution is a discrete probability distribution that is used to obtain the probability of observing exactly k number of successes in a sequence of n trials, with the probability of success for all single trials of p.
The shape of the binomial distribution is dependent on the values of n and p, as illustrated in the following diagrams:
The normal distribution is a continuous probability distribution whose shape is dependent on the mean, m, and variance, σ.
Since a binomial variate, B(n,p), is a sum of n independent, identically distributed Bernoulli variables with parameter p, it follows that by the central limit theorem it can be approximated by the normal distribution with mean n p and variance n p 1−p, provided that both n p >5 and n 1−p > 5.
In the following example, adjust the binomial probability mass function by moving the sliders for the values of n and p. The red line shows the probability density function for the normal distribution.
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