a. We can see that the t distribution is a mixture of normals using the following argument:

where T" is a t random variable with v degrees of freedom. Using the Fundamental Theorem of Calculus and interpreting P{xt = ux) as a pdf, we obtain

a scale mixture of normals. Verify this formula by direct integration.
b. A similar formula holds for the F distribution; that is, it can be written as a mixture of chi squareds. If F^ is an F random variable with 1 and v degrees of freedom, then we can write

where fu(y) is a x2v pdf. Use the Fundamental Theorem of Calculus to obtain an integral expression for the pdf of Fi,v1 and show that the integral equals the pdf.
c. Verify that the generalization of part (b),

is valid for all integers m > 1.

'p(nst) = P(7.sj..7. 0 e"2