5. If X gamma(alpha, beta), then Y=1/X has the inverted gamma distribution.
(a) Derive the form of the pdf;
(b) verify that it is a pdf;
(c) and calculate the mean and variance
6. Let X be a positive random variable, i.e. P(X greater or less than 0) = 0. Prove that E(ln(1/x) greater of less than ln(1/E(X)).
This solution provides step-by-step calculations for inverted gamma distributions and natural logarithms.