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Derivations of Student’s-T and the F Distributions. Student’s-T Distribution (P. 1). Student’s T-Distribution (P. 2). Step 1: Fix V=v and write f ( z|v ) =f ( z ) (by independence) Step 2: Let T = h ( Z ) (and Z= h -1 ( T )) and obtain f ( t|v ) by method of transformations:

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student s t distribution p 2
Student’s T-Distribution (P. 2)
  • Step 1: Fix V=v and write f(z|v)=f(z) (by independence)
  • Step 2: Let T = h(Z) (and Z=h-1(T)) and obtain f(t|v)by method of transformations:

fT(t|v) = fZ(h-1(t)|v)|dZ/dT|

  • Step 3: Obtain joint distribution of T, V :

fT,V(t,v) = fT(t|v) fV(v)

  • Step 4: Obtain marginal distribution of T by integrating the joint density over V and putting in form:
student s t distribution p 3
Student’s-T Distribution (P. 3)

Conditional Distribution of T|V=v and Marginal Distribution of V

student s t distribution p 4
Student’s-T Distribution (P. 4)

Marginal Distribution of T (integrating out V) (Continued below)

student s t distribution p 5
Student’s-T Distribution (P. 5)

Marginal Distrbution of T

f distribution p 2
F-Distribution (P.2)
  • Step 1: Fix W=w f(v|w) = f(v) (independence)
  • Step 2: Let F=h(V) and V=h-1(F) and obtain fF(f|w) by method of transformations:

fF(f|w) = fV(h-1(f)|w) |dV/dF|

  • Step 3: Obtain the joint distribution of F and W

fF,W(f,w) = fF(f|w) fW(w)

  • Step 4: Obtain marginal distribution of F by integrating joint density over W and putting in form:
f distribution p 3
F-Distribution (P. 3)

Conditional Distribution of F|W=w

f distribution p 4
F-Distribution (P. 4)

Marginal Distribution of W, Joint Distribution of F,W

f distribution p 5
F-Distribution (P. 5)

Marginal Distribution of F