Continuous probability distribution is that type of distribution that deals with continuous type of data or random variables. The continuous random variables deal with different kinds of continuous probability distribution.
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There are different continuous probability distributions.
A normal distribution is a continuous probability distribution 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).
This type of continuous probability distribution 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 continuous probability distribution.
This continuous probability distribution finds a large number of applications in Statistical Quality Control.
This type of continuous probability distribution 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 to this type of continuous probability distribution.
Even if the variable is not following this type of continuous probability distribution, then it can be transformed into this type of continuous probability distribution.
A gamma distribution is a continuous probability distribution 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 of continuous probability distribution 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 this continuous probability distribution.
A beta distribution of the first kind is a continuous probability 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 continuous probability 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 continuous probability 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 a continuous probability distribution with no parameter. The reason that there is no parameter in this type of continuous probability distribution is because this continuous probability distribution is standardized in nature. Thus, this continuous probability distribution does not have any parameters. Its continuous probability distribution is given by the following:
f(x)= 0.5 exp (- )
A weibul distribution is a continuous probability 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 continuous probability distribution with parameter a and ß. This type of continuous probability distribution is used widely as a growth function in population and other demographic studies. This type of continuous probability distribution is considered to be the mixture of the extreme values of the distributions.
A Cauchy distribution is a continuous probability distribution with parameter ‘l’ > 0 and ‘µ.’ This type of continuous probability distribution 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 of continuous probability distribution follows the additive property as stated above. The type of continuous probability distribution plays a role in providing counter examples.
