1. Derivations of Student’s-T and the F Distributions

2. Student’s-T Distribution (P. 1)

3. 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:

4. Student’s-T Distribution (P. 3)
Conditional Distribution of T|V=v and Marginal Distribution of V

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

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

8. F-Distribution (P. 1)

9. 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:

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

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

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