Discrete Random Distribution

Discrete Random Distribution

Ethara

Most common discrete random distributions are Bernoulli, binomial, multinomial, geometric and Poisson distributions. Let’s elaborate on them one by one.

Bernoulli Distribution

Suppose a random variable X can be either 0 or 1, and its probability distribution is,

where 0 < p < 1, then,

Apparently, the mean and the variance of Bernoulli distribution are p and p(1-p) respectively.

Binomial Distribution

In Bernoulli experiment, there are only two results that can possibly happen. Moreover, Binomial distribution is thought of as n times Bernoulli experiment, where X denotes the number of times that event A takes place. Thus,

where k = 0, 1, 2,…, n.

The mean and the variance of Binomial distribution are np and np(1-p) respectively.

Comparing adjacent terms yields the maximum likelihood number of times,

Multinomial Distribution

The major difference from Binomial distribution is that there are k results (k > 2), denoted by A₁, A₂, … A_k, which can possibly happen in a Bernoulli experiment.

Suppose

Obviously,

In n times Bernoulli experiment, let X₁, X₂, …, X_k be the number of times that A₁, A₂, …, A_k take place respectively. Then what is the probability that X₁ =n₁, X₂ =n₂, …, X_k =n_k? (n₁ + n₂ + … + n_k = n)

Alternatively, for a polynomial  

the general term of its expansion is exactly,

In particular, if 

,

then we have, 

Note that (*) = 1, we obtain the following equation, 

Geometric Distribution

In an infinite times Bernoulli experiment, P(A) = p, let X denote the number of experiments taken after A takes place for the very first time, then,

where k = 1, 2, ….

What is the mean of geometric distribution?

This is an infinite series. Let x = 1 – p, the sum function is,

Take integrals with regard to x on both sides,

Take derivatives with regard to x on both sides,

Thus,

Further, we can solve for the variance using

Poisson Distribution

Poisson distribution is the limiting case of Binomial distribution by,

where k = 0, 1, 2, ….

The mean and the variance are both lambda.