Implementing Statistical Distributions

Aim: Implementing Statistical Distributions

Discrete distributions

a)      Bernoulli distribution

b)     Binomial distribution

c)      Poisson distribution

d)     Multinomial distribution

e)      Hyper geometric distribution

Continuous distributions

a)      Normal distribution

b)     Lognormal distribution

c)      Gamma distribution

d)     Exponential distribution

e)      Beta distribution

Concept:

A discrete probability distribution shall be understood as a probability distribution characterized by a probability mass function. Thus, the distribution of a random variable X is discrete, andX is then called a discrete random variable, if

as u runs through the set of all possible values of X. It follows that such a random variable can assume only a finite or countably infinite number of values. For the number of potential values to be countably infinite even though their probabilities sum to 1 requires that the probabilities decline to zero fast enough: for example, if  for n = 1, 2, ..., we have the sum of probabilities 1/2 + 1/4 + 1/8 + ... = 1.

Among the most well-known discrete probability distributions that are used for statistical modeling are the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution. In addition, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.

A continuous probability distribution is a probability distribution that has a probability density function. Mathematicians also call such a distribution absolutely continuous, since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure λ. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.