# Derivations of Student’s-T and the F Distributions

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

Student’s-T Distribution (P. 4)
Marginal Distribution of T (integrating out V) (Continued below)

Student’s-T Distribution (P. 5)
Marginal Distrbution of T

F-Distribution (P. 1)

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:

Conditional Distribution of F|W=w
F-Distribution (P. 3) Conditional Distribution of F|W=w

Marginal Distribution of W, Joint Distribution of F,W
F-Distribution (P. 4) Marginal Distribution of W, Joint Distribution of F,W

Marginal Distribution of F
F-Distribution (P. 5) Marginal Distribution of F