Chi-Square Distribution

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An Introduction to the Chi-Square Distribution
5:29 — by jbstatistics

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Key Questions

  • by definition a chi-square distribution is the square of a standard normal distribution.
    suppose a population follows normal distribution with parameters as mean and variance, will be symmetric in nature and like that a standard normal variate will follow the same property. thus chi-square will be square of that standard normal variate, and will follow the same properties.

    it will have "n" degrees of freedom,as degrees of freedom is the difference between numbers of observation and limitation or estimated parameters. thus here as there is no estimated parameters are give the chi-square will have "n" degrees of freedom.
    view the above link to have a further clear perspective.

  • Each chi-square distribution depends on the degrees of freedom parameter, #k#, specified so it is a family of distributions.

    The probability distribution for a random variable #x# that follows a chi-square distribution that has #k# degrees of freedom is

    #f(x | k) = 1/(2^(k/2) Gamma(k/2)) x^(k/2 -1 ) e^(-x/2)# for #x>0# and #k=1, 2, 3, ...#

    Some examples of chi-square distribution shapes follows:

    enter image source here

  • The degrees of freedom

    The shape of the chi-squared curve is the cumulative density function or probability density function, depending of what you want, of a random variable #X# with a chi squared distribution with k degrees of freedom(#chi_(k)^2 #).The only factors that can interfere with the shape of those graphs are the parameters taken by the distribution, the chi-squared distribution only takes one parameter, that is, the degrees of freedom, so it is the only one that can interfere in its shape.

    There is a twist thought , it is possible to have a Non-central chi-squared distribution , this is common in some areas of statistics and in that case you have an extra parameter

    Here are some graphs of the central chi-squared distribtution:

    If you having trouble understanding what the chi-squared means, maybe if you use the relation it has with other variables to enlighten you the relation it has with other variables to enlighten yourself