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Student's t Distribution

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Distribution

A random variable ~Y is said to have the #~{student's t distribution} with ~k degrees of freedom if it has the probability density function:

f( ~y ) _ = _ fract{&Gamma.( &hlf.(~k+1) )( 1 + ~y^2/~k )^{-&hlf.(~k+1)}, &Gamma.( &hlf.~k ) &sqrt.${~k&pi.}}

where &Gamma.( &alpha. ) is the gamma function .

We write _ ~Y _ ~ _ t ( ~k ) .

Note that as _ &sqrt.${&pi.} = &Gamma.( &hlf. ) , _ so the density function can be rewritten:

f( ~y ) _ = _ fract{&Gamma.( &hlf.(~k+1) ) ( 1 + ~y^2/~k )^{-&hlf.(~k+1)}, &Gamma.( &hlf.~k )&Gamma.( &hlf. ) &sqrt.${~k}}

_ _ _ _ _ _ _ = _ fract{1,{&Beta.( &hlf.~k , &hlf. )}} ( ~k )^{-&hlf.} ( 1 + ~y^2/~k )^{-&hlf.(~k+1)},

where _ &Beta.( ~a , ~b ) is the beta function.