Continuous Probability Distribution

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Continuous probability distribution is a type of distribution that deals with continuous types of data or random variables. The continuous random variables deal with different kinds of distributions.

There are different types of continuous probability distributions.

A normal distribution is one with parameters µ ( called the mean) and s2 (called the variance) that have a range of -8 to +8. Its continuous probability distribution is given by the following:

f(x;µ, s)= (1/ s p) exp(-0.5 (x-µ)2/ s2).

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This type plays a crucial role in statistical theory for several reasons. Most of the distributions, like binomial, poisson and hyper geometric distributions are approximated with the help of this distribution.

This continuous probability distribution finds a large number of applications in Statistical Quality Control.

This type is used widely in the study of large sample theory where normality is involved. Sample statistics can be best studied with the help of the curves of this type of continuous probability distribution.

The overall theory of significance tests (like t test, F test, etc.) are entirely based upon the fundamental assumption that the parent population belongs.

Even if the variable is not following, then it can be transformed into this type of continuous probability distribution.

A gamma distribution is one with the parameter ‘d>0’that has a range of 0 to 8. Its continuous probability distribution is given by the following:

f(x)= exp(-x) xd-1/

This type has a property called the additive property. This property states that the sum of the independent variates of this continuous probability distribution is equal to the variate of it.

A beta distribution of the first kind is a distribution with the parameters µ>0 and v>0 that has the range of 0 to 1. Its continuous probability distribution is given by the following:

f(x)= (1/B(µ,v)) xµ-1 (1-x)v-1

A beta distribution of the second kind is a distribution with the parameters µ>0 and v>0 that has the range of 0 to 8. Its continuous probability distribution is given by the following:

f(x)= (1/B(µ,v)) xµ-1 (1+x)v+µ

An exponential distribution is a distribution with the parameter ‘c’ >0 that has the range of 0 to 8. Its continuous probability distribution is given by the following:

f(x,c)= c exp(-cx)

A standard laplace or double exponential distribution is one with no parameter. The reason that there is no parameter in this type is because this continuous probability distribution is standardized in nature. Thus, it does not have any parameters. Its continuous probability distribution is given by the following:

f(x)= 0.5 exp (- )

A weibul distribution is a distribution with three parameters c(>0), a(>0) and µ that has the range of µ to 8. Its continuous probability distribution is given by the following:

f(x;c,a,µ) = (c (x-µ/a)c-1)/ a exp (-(x-µ/a)c)

A logistic distribution is a distribution with parameter a and ß. This type is used widely as a growth function in population and other demographic studies. It is considered to be the mixture of the extreme values of the distributions.

A Cauchy distribution is a distribution with parameter ‘l’ > 0 and ‘µ.’ This type has the range of -8 to +8. The continuous probability distribution is given by the following:

f(x)= l/p(l2+(x-µ)2)

This type follows the additive property as stated above. It plays a role in providing counter examples.

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