The Student s t and F Distributions Page 1

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1 The Suden s and F Disribuions Page The Fundamenal Transformaion formula for wo random variables: Consider wo random variables wih join probabiliy disribuion funcion f (, ) simulaneously ake on values in region D of he x-y plane is given by f f ( xy, ) y x Al Lehnen Madison Area Technical College 3/8/ D xy. The probabiliy ha x and y x, y dydx. Thus, one can inerpre ΔΔ as he approximae probabiliy of being in a small recangle cenered a (x, y) and his approximaion becomes increasingly accurae as he dimensions of he recangle become smaller and smaller. Suppose now we u g x, y v h x, y. We suppose ha a leas in he viciniy ransform o wo new random variables u and v, wih and of (x, y) ha his ransformaion is one o one so ha inverse ransformaions x Guv (, ) and y H( u, v) D v v u g x, y mulivariae calculus is shown ha exis. In f xydydx, f( Guv,, H uv, ) abs dvduwhere D is he image of D he boundary of D in he u-v plane goen by using he direc ransformaion and v h( x, y) u abs v. is he absolue value of he Jacobian deerminan which scales areas in he u-v plane ino he v f uv has corresponding areas in he x-y plane. Since he probabiliy densiy funcion of he new random variables, (, ) he propery ha f ( u, v) dvdu f ( x, y) dydx, we conclude ha D D f( u, v) f( G( u, v), H( u, v) ) abs. () v v This is he fundamenal ransformaion formula for probabiliy densiy funcions of wo random variables. x μ When doing saisical inference on he mean based on random samples he saisic, x, is he closes sx n approximaion based on known informaion o he z score sampled from he normal disribuion of sample x μ means N ( μx, x ). Now, x x μx. If he paren populaion is he normal disribuion N ( μx, x ), sx n x n sx ( n) n x x μ hen z x sx n is a random score from N(,and ) is an independen random score from x n χ n, a chi x squared disribuion wih n degrees of freedom. This moivaes he join probabiliy disribuion of defined as he z raio where z and are independen random variables, z from a sandard normal disribuion and from a chi squared disribuion wih degrees of freedom. From he assumpion of independen random variables he join probabiliy densiy funcion is given below by equaion ().

2 The Suden s and F Disribuions Page f z, ( z ) z / / e e > π <, Γ Now make he ransformaion o he new variables and. z if if z This has he inverse equaion, z, so he Jacobian deerminan is given by z Thus from equaions () and (), / / e e if > f,, fz,, π v Γ (3) if < The marginal probabiliy disribuion of is given by (). Making he subsiuion u ( /)( ) ( )( ) ( ) / + f() f, (, ) d e d π vγ + wih u +, he inegral becomes ( + ) ( + ) + u () ( ) f e u d π vγ + Γ + f () + (4) π vγ This is he Suden s disribuion wih degrees of freedom. The gamma funcion is defined for p > by x p Γ p e x dx. Inegraing by pars yields he fundamenal recurrence, ( p) Γ( p) For p a posiive ineger x p x p x p ( p) e x dx e x ( p ) e x dx π ( p ) Γ p p! For large u he gamma funcion is well approximaed by Sirling s formula (hp://faculy.macmadison.edu/alehnen/engineeringsas/sirling%7s_formula.pdf ). Al Lehnen Madison Area Technical College 3/8/ Γ +

3 The Suden s and F Disribuions Page 3 Applying his o he disribuion for large, f () u x u u ( u) e x Γ dx π ( u) e, e ( + ) exp ln v e + π ( ) ( ) ( + ) exp ln π v + e + exp ln ln π ln + exp A( ) π x 3 ln + x x + O x, he funcion A( ) Using he Maclaurin series for he naural logarihm: follows: So we have ha f () ( ) ( + ) can be expanded as A + O v O v O v + + ( ) ( ) ( + O v + + O v + O v ) + O( v ) exp as. Thus, a large degrees of freedom a disribuion approaches a π sandard normal disribuion. Graphs of he disribuion for various degrees of freedom are shown in Figure. Because pdf of he disribuion is an even funcion of for any number of degrees of freedom, μ. Looking a + + ( + ) Γ + Γ Equaion (4) we see ha for large ( + f ) () +, hus πvγ πvγ if and only if he improper inegral d will converge converges. This requires ha he degrees of freedom be hree or greaer. For one and wo degrees of freedom, he disribuion has oo much probabiliy in i ails, so ha he sandard deviaion is infinie! Al Lehnen Madison Area Technical College 3/8/

4 The Suden s and F Disribuions Page 4 Figure An easy way o calculae for more han wo degrees of freedom is o use equaion () and he expecaion of in a chi squared disribuion wih more han wo degrees of freedom. Γ u e d e u du Γ Γ Γ Γ Γ Now, z z, z, (, ) (, ) f z ddz f z ddz z z So he variance of a disribuion wih more han wo degrees of freedom is. Al Lehnen Madison Area Technical College 3/8/

5 The Suden s and F Disribuions Page 5 Summary for a Suden s disribuion + Γ ( + ) Probabiliy Densiy Funcion f () + π vγ Mean μ Sandard Deviaion for 3 When doing saisical inference on he raio of wo populaion variances based on independen random samples one calculaes he F saisic, ( n ) s s F. If boh populaions are normally disribued, he raios s ( n ) s and are disribued like chi squared disribuions wih n and n degrees of freedom respecively. This moivaes he join probabiliy disribuion of F defined as he raio F where and are independen random variables each from a chi squared disribuion wih wih n and n degrees of freedom respecively. From he assumpion of independence he join probabiliy densiy funcion is given by / / e e if > and > f,, Γ Γ (6) elsewhere We make he ransformaion from and o F and, F and which has he inverse ransformaion F F F and. Thus, he Jacobian deerminan is given by. F Thus from equaions () and (6), F/ F / e e F F,,,, F Γ Γ elsewhere f F f if > and > So ha he marginal probabiliy disribuion of F is given by he following: Al Lehnen Madison Area Technical College 3/8/

6 The Suden s and F Disribuions Page 6 F ( F/ ) / ( ) ff,, F f F, F, d e + + d ( + ) Γ Γ F ( + ) ( + ) u ( + ) + F e u du ( + ) Γ Γ F ( + ) F + + Γ Γ Γ + Γ F ff,, F (7) + Γ Γ + F The F disribuion is characerized by wo degrees of freedom which goes wih he numeraor variance and which + Γ + goes wih he denominaor variance. For large F, ff,, F F so ha for any valid Γ Γ denominaor degree of freedom he oal area under he disribuion is finie (in fac i s one), bu + Γ + Γ Ff F,, F F and F f F,, F F so F will only exis for Γ Γ Γ Γ 3and F will only exis for 5. For small values of he denominaor degrees of freedom he probabiliy of large F scores is no small enough for he mean or sandard deviaion o exis. The momens, when hey do exis, are calculaed mos easily using he join chi squared disribuion of equaion (6). F F f d d f d d,,,, f χ d f d χ Al Lehnen Madison Area Technical College 3/8/

7 The Suden s and F Disribuions Page 7 For a chi squared disribuion F,, F f, d d f, d d fχ ( ) d f d χ + x u + χ x x f x dx x x e dx e u du Γ Γ Γ Γ ( ) Γ Γ x u 3 x x f χ x dx x x e dx e u du Γ Γ Γ Γ ( v)( v4) 4Γ 4 4 Γ So, ( + ) F ( v + v ) v 4 4 expression for he variance of an F disribuion. v ( ) v( + ) ( 4) F F F F F v From he sandard formula we obain he following ( + ) v 4 ( ) v [ 4 4 ] Graphs of he F disribuion are shown in Figure 5 for several differen choices of numeraor and denominaor degrees of freedom. Like he chi squared disribuions from which i was derived, he F disribuion is no very robus from deparures of he wo paren populaions from normaliy. Al Lehnen Madison Area Technical College 3/8/

8 The Suden s and F Disribuions Page 8 Figure One imporan ransformaion on he F disribuion concerns calculaing is criical values. F α (, ) score in an F disribuion wih numeraor degrees of freedom and probabiliy o he righ equal oα. More precisely, + Γ F (, ) (, ) α Fα F, F, + Γ Γ + F is defined as he denominaor degrees of freedom wih a f F df df α. Le G, he probabiliy ha G< K means F > K.Thus, he pdf for G is given by F d fg( G) f F,, F df f F,, G G f F,, G G dg G Al Lehnen Madison Area Technical College 3/8/

9 The Suden s and F Disribuions Page G Γ d Γ G fg( G) f G F,, F df dg Γ Γ Γ Γ + + G G G G G + Γ G Γ + + Γ Γ Γ Γ + G + G ff,, ( G) Tha is, swiching he numeraor and denominaor degrees of freedom gives he pdf of he reciprocal. Thus, F α (, ) which is he F score wih a probabiliy ofα o he lef implies ha G has a probabiliy ofα o he righ. F Thus, F α (, ) F,. This symmery is useful in calculaing lef ail probabiliies of F disribuions. α Summary for an F disribuion Probabiliy Densiy Funcion f ( F) Mean Sandard Deviaion + Γ F F,, + Γ Γ + F μ F F for ( + ) ( ) F for 5 4 Al Lehnen Madison Area Technical College 3/8/