The chi-squared distribution is widely used in statistics. It is a special case of [Gamma distributions] and is also related to the [Gaussian distribution], in that the sum of squares of identically and independently distributed (iid) gaussian variables follow a chi-squared distribution.

For an overview, see the Wikipedia article [http://en.wikipedia.org/wiki/Chi-square_distribution] and Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_940.htm].

[EKB] This is an implementation of the chi-squared distribution, providing cdf-chisquare, pdf-chisquare, and random-chisquare. It uses code for [Gamma distributions] and the [Simple Test] code. (I used Simple Test to implement this using [test-driven development].) To test the random-number generation, there is test code at the end to compare a histogram of the random variates against the theoretical pdf-chisq.

    # Implements the (central) chi-squared distribution
    # as a special case of the gamma distribution

    source simpletest.tcl
    source gammadist.tcl

    namespace eval chisq {
        
        # Values were produced from R ver. 2.4.1
        
        st::addproc cdf-chisquare {
            df  ;# Degrees of freedom
            x   ;# chisquared value; find prob that < this
        } where {
            {{2 3.5} ~ 0.826226056549555}
            {{5 2.2} ~ 0.179164030785504}
            {{5 100} ~ 1.0}
            {{3.9 4.2} ~ 0.634682741547709}
            {{1 2.0} ~ 0.842700792949715}
            {{3 -2.0} -> 0.0}
        }
        
        st::addproc pdf-chisquare {
            df  ;# Degrees of freedom
            x   ;# chisquared value; eval prob dens at this value
        } where {
            {{3 1.75} ~ 0.219999360547348}
            {{10 2.9} ~ 0.0216024880121444}
            {{4 17.45} ~ 0.000708787557977144}
            {{2.5 1.8} ~ 0.218446210041615}
            {{1 0.0} -> Inf}
            {{5 -1.5} -> 0.0}
        }
    }

    proc chisq::cdf-chisquare {df x} {
        set alpha [expr {0.5 * $df}]
        set beta 0.5
        
        if {$x < 0.0} {return 0.0}
        
        cdf-gamma $alpha $beta $x
    }

    proc chisq::pdf-chisquare {df x} {
        set alpha [expr {0.5 * $df}]
        set beta 0.5
        
        if {$x < 0.0} {return 0.0}
        
        pdf-gamma $alpha $beta $x
    }

    # Prodce "number" random deviates distributed according to a chisq dist
    proc chisq::random-chisquare {df number} {
        set alpha [expr {0.5 * $df}]
        set beta 0.5
        
        random-gamma $alpha $beta $number    
    }

    console show
    namespace eval chisq {
        st::print
    }

    ### Test random deviates

    ##########################################################################################
    ##
    ## TESTING
    ##
    ## Can test pdf & cdf by running in a console. For random numbers, generate histograms:
    ##
    ##########################################################################################
    canvas .c
    pack .c -side top
    frame .f
    pack .f -side bottom
    label .f.dfl -text "Degrees of freedom: "
    entry .f.dfe -textvariable df
    pack .f.dfl -side left
    pack .f.dfe -side left
    button .f.run -text "Run" -command runtest
    pack .f.run -side left

    proc runtest {} {
        set vals [chisq::random-chisquare $::df 5000]
        set remainder 5000
        for {set x 0.0} {$x < 20.0} {set x [expr {$x + 0.25}]} {
            lappend bins $x
            set distval [chisq::pdf-chisquare $::df $x]
            # Total trials (5000) * step (0.25) * value
            set distval [expr {int(5000 * 0.25 * $distval)}]
            lappend distcounts $distval
        }
        # Assume none are left
        lappend distcounts 0.0
        set bincounts [::math::statistics::histogram $bins $vals]
        .c delete hist
        .c delete dist
        math::statistics::plot-scale .c 0 20 0 [math::statistics::max $bincounts]
        math::statistics::plot-histogram .c $bincounts $bins hist
        math::statistics::plot-histogram .c $distcounts $bins dist
        .c itemconfigure dist -fill {} -outline green
    }

    set df 3
    runtest


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