Intoduction to Mathematical Statistics Robet V. Hogg Joeseph McKean Allen T. Caig Seventh Edition
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Define a new andom vaiable T by witing T W V/. The change-of-vaiable technique is used to obtain the pdf g (t) oft. The equations t w v/ and u v define a tansfomation that maps S {(w, v) : <w<, <v< } one-to-one and onto T {(t, u) : < t <, < u < }. Since w t u/, v u, the absolute value of the Jacobian of the tansfomation is J u/. Accodingly, the joint pdf of T and U V is given by ( ) t u g(t, u) h,u J { ( )] u u 2 + t2 elsewhee. 2πΓ(/2)2 /2 u/2 exp The maginal pdf of T is then t <, <u< g (t) g(t, u) du 2πΓ(/2)2 /2 u(+)/2 exp u ( )] + t2 du. 2 In this integal let z u + (t 2 /)]/2, and it is seen that g (t) ( ) (+)/2 ( ) 2z 2πΓ(/2)2 /2 +t 2 e z 2 / +t 2 / Γ( +)/2], <t<. (6.) πγ(/2) ( + t 2 /)(+)/2 Thus, if W is N(, ), if V is χ 2 (), and if W and V ae independent, then T dz W V/ (6.2) has the immediately peceding pdf g (t). The distibution of the andom vaiable T is usually called a t-distibution. It should be obseved that a t-distibution is completely detemined by the paamete, the numbe of degees of feedom of the andom vaiable that has the chi-squae distibution. Some appoximate values of P (T t) t g (w) dw 92
fo selected values of and t can be found in Table IV in Appendix: Tables of Distibutions. Note that the last line of the this table, which is labeled, contains the N(, ) citical values. This is because as the degees of feedom appoach, the t-distibution conveges to the N(, ) distibution. The R compute package can also be used to obtain citical values as well as pobabilities concening the t-distibution. Fo instance, the command qt(.975,5) etuns the 97.5th pecentile of the t-distibution with 5 degees of feedom; the command pt(2.,5) etuns the pobability that a t-distibuted andom vaiable with 5 degees of feedom is less that 2.; and the command dt(2.,5) etuns the value of the pdf of this distibution at 2.. Remak 6.. This distibution was fist discoveed by W. S. Gosset when he was woking fo an Iish bewey. Gosset published unde the pseudonym Student. Thus this distibution is often known as Student s t-distibution. Example 6. (Mean and Vaiance of the t-distibution). Let the andom vaiable T have a t-distibution with degees of feedom. Then, as in (6.2), we can wite T W (V/) /2,wheeW has a N(, ) distibution, V has a χ 2 () distibution, and W and V ae independent andom vaiables. Independence of W and V and expession (3.4), povided (/2) (k/2) > (i.e., k<), implies the following: ( ) ] k/2 (V ) ] k/2 V E(T k ) E W k E(W k )E (6.3) E(W k ) 2 k/2 Γ ( 2 ) k 2 Γ ( ) if k<. (6.4) 2 k/2 Fo the mean of T,usek. Because E(W ), as long as the degees of feedom of T exceed, the mean of T is. Fo the vaiance, use k 2. In this case the condition >kbecomes >2. Since E(W 2 ), by expession (6.4), the vaiance of T is given by Va(T )E(T 2 ) 2. (6.5) Theefoe, a t-distibution with >2degees of feedom has a mean of and a vaiance of /( 2). 6.2 The F -distibution Next conside two independent chi-squae andom vaiables U and V having and 2 degees of feedom, espectively. The joint pdf h(u, v) ofu and V is then { u /2 v 2/2 e (u+v)/2 <u,v< h(u, v) Γ( /2)Γ( 2/2)2 ( + 2 )/2 elsewhee. We define the new andom vaiable W U/ V/ 2 93
and we popose finding the pdf g (w) ofw. The equations w u/ v/ 2, z v, define a one-to-one tansfomation that maps the set S {(u, v) :<u<, < v < } onto the set T {(w, z) : < w <, < z < }. Since u ( / 2 )zw, v z, the absolute value of the Jacobian of the tansfomation is J ( / 2 )z. The joint pdf g(w, z) of the andom vaiables W and Z V is then g(w, z) Γ( /2)Γ( 2 /2)2 (+2)/2 ( zw 2 ) 2 2 z 2 2 2 exp z ( )] w z +, 2 2 2 povided that (w, z) T, and zeo elsewhee. The maginal pdf g (w) ofw is then g (w) g(w, z) dz ( / 2 ) /2 (w) /2 Γ( /2)Γ( 2 /2)2 (+2)/2 z(+2)/2 exp z ( )] w + dz. 2 2 If we change the vaiable of integation by witing it can be seen that g (w) y z 2 ( ) w +, 2 ( / 2 ) /2 (w) /2 ( Γ( /2)Γ( 2 /2)2 ( ) (+2)/2 2 dy w/ 2 + { Γ(+ 2)/2]( / 2) /2 2y w/ 2 + (w) /2 (+ w/ 2) ( + 2 )/2 ) (+ 2)/2 e y Γ( /2)Γ( 2/2) <w< elsewhee. (6.6) Accodingly, if U and V ae independent chi-squae vaiables with and 2 degees of feedom, espectively, then W U/ V/ 2 has the immediately peceding pdf g(w). The distibution of this andom vaiable is usually called an F -distibution; and we often call the atio, which we have denoted by W, F.Thatis, F U/ V/ 2. (6.7) 94
It should be obseved that an F -distibution is completely detemined by the two paametes and 2. Fo selected values of, 2,andb, Table V in Appendix: Tables of Distibutions gives some appoximate values of P (F b) b g (w) dw. The R package can also be used to find citical values and pobabilities fo F - distibuted andom vaiables. Suppose we want the.25 uppe citical point fo an F andom vaiable with a and b degees of feedom. This can be obtained by the command qf(.975,a,b). Also, the pobability that this F -distibuted andom vaiable is less than x is etuned by the command pf(x,a,b), while the command df(x,a,b) etuns the value of its pdf at x. Example 6.2 (Moments of F -Distibutions). Let F have an F -distibution with and 2 degees of feedom. Then, as in expession (6.7), we can wite F ( 2 / )(U/V ), whee U and V ae independent χ 2 andom vaiables with and 2 degees of feedom, espectively. Hence, fo the kth moment of F, by independence we have E ( F k) ( ) k 2 E ( U k) E ( V k), povided, of couse, that both expectations on the ight side exist. By Theoem 3., because k> ( /2) is always tue, the fist expectation always exists. The second expectation, howeve, exists if 2 > 2k; i.e., the denominato degees of feedom must exceed twice k. Assuming this is tue, it follows fom (3.4) that the mean of F is given by E(F ) ( 2 2 Γ 22 ) Γ ( ) 22 2 2 2. (6.8) If 2 is lage, then E(F ) is about. In Execise 6.6, a geneal expession fo E(F k ) is deived. 6.3 Student s Theoem Ou final note in this section concens an impotant esult fo infeence fo nomal andom vaiables. It is a coollay to the t-distibution deived above and is often efeed to as Student s Theoem. Theoem 6.. Let X,...,X n be iid andom vaiables each having a nomal distibution with mean μ and vaiance σ 2. Define the andom vaiables X n n i X i and S 2 n n i (X i X) 2. Then ( ) (a) X has a N μ, σ2 n distibution. 95
(b) X and S 2 ae independent. (c) (n )S 2 /σ 2 has a χ 2 (n ) distibution. (d) The andom vaiable T X μ S/ n has a Student t-distibution with n degees of feedom. (6.9) Poof: Note that we have poved pat (a) in Coollay 4.2. Let X (X,...,X n ). Because X,...,X n ae iid N(μ, σ 2 ) andom vaiables, X has a multivaiate nomal distibution N(μ,σ 2 I), whee denotes a vecto whose components ae all. Let v (/n,...,/n) (/n). Note that X v X. Define the andom vecto Y by Y (X X,...,X n X). Conside the following tansfomation: W X Y ] v I v ] X. (6.) Because W is a linea tansfomation of multivaiate nomal andom vecto, by Theoem 5. it has a multivaiate nomal distibution with mean ] ] v E W] μ I v μ, (6.) n whee n denotes a vecto whose components ae all, and covaiance matix ] ] v Σ I v σ 2 v I I v ] σ 2 n n n I v. (6.2) Because X is the fist component of W, we can also obtain pat (a) by Theoem 5.. Next, because the covaiances ae, X is independent of Y. But S 2 (n ) Y Y. Hence, X is independent of S 2, also. Thus pat (b) is tue. Conside the andom vaiable n ( ) 2 Xi μ V. σ i Each tem in this sum is the squae of a N(, ) andom vaiable and, hence, has a χ 2 () distibution (Theoem 4.). Because the summands ae independent, it follows fom Coollay 3. that V is a χ 2 (n) andom vaiable. Note the following identity: n ( ) 2 (Xi X)+(X μ) V σ i n ( ) 2 ( ) 2 Xi X X μ + σ σ/ n i ( ) 2 (n )S2 X μ σ 2 + σ/. (6.3) n 96