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Derivations of Student’s-T and the F Distributions

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Student’s-T Distribution (P. 1)

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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: f T (t|v) = f Z (h -1 (t)|v)|dZ/dT| Step 3: Obtain joint distribution of T, V : f T,V (t,v) = f T (t|v) f V (v) Step 4: Obtain marginal distribution of T by integrating the joint density over V and putting in form:

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

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

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

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F-Distribution (P. 1)

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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 f F (f|w) by method of transformations: f F (f|w) = f V (h -1 (f)|w) |dV/dF| Step 3: Obtain the joint distribution of F and W f F,W (f,w) = f F (f|w) f W (w) Step 4: Obtain marginal distribution of F by integrating joint density over W and putting in form:

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

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

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

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