Continuous Probability distribution

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.

Chi-square

Chi square is defined as the square of the standard normal variable. There are certain chi square tests and they are discussed below in a detailed manner.

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A cross tabulation is also a kind of chi square test that is used by the researcher in order to test the statistical significance of the correlation that is observed in the study. The chi square test is used by the researcher to determine the strength of the association in the objects under study.

The researcher should note that the greater the difference between the observed value of the cell frequency and the expected value of the cell frequency, the larger the value of the statistic of the chi square. This means that the difference of the observed value and the expected value in the chi square test is directly proportional to the value of the chi square statistic in the chi square test.

To determine the association or the correlation between the two variables that exist in the chi square test, the probability that is computed for obtaining the value of the chi square must be larger or greater, or must have a higher value than the one obtained, which is computed from the chi square test of cross tabulation.

Another popular chi square test is the goodness of fit test. This goodness of fit in the chi square test helps the researcher to understand whether or not the sample that is collected from some population belongs to some specific distribution. This chi square test is basically applicable in cases where the discrete type of probability distributions is involved, like Poisson distribution, binomial distribution, etc. This chi square test is an alternative to the non parametric type of test, called the Kolmogorov Smirnov goodness of fit test.

The null hypothesis that the researcher assumes in this chi square test is that the drawn data from the population follows the distribution. The definition of the statistic used in the chi square test is the same, which is the sum of the square of the deviation between the observed and the expected frequency that is divided by the expected frequency. An important point related to the validity of this type of chi square test is that the expected number of cell frequencies should be less than five.

Researchers generally assume certain assumptions in the chi square test, and on the basis of those assumptions, only the chi square test is carried out.

The first assumption in the chi square test is that the sampling of the data is collected by the process of random sampling from the population.

A sample size that is sufficiently large is assumed in the chi square test. The chi square test that is conducted on the sample of a smaller size results in the drawing of an inaccurate inference about the data. If the researcher conducts the chi square test on a small sample size, then it may happen that the researcher might end up committing a Type II error.

As in all other significant tests, it is assumed that in the chi square test, the observations are always independent of each other.

The last assumption that is made in the chi square test is that the observations in the sample must acquire the same fundamental distribution.