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Inverted Gamma Distribution

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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)).

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This solution provides step-by-step calculations for inverted gamma distributions and natural logarithms.

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Question 5

a. We know that the distribution of X is such that:

We now find the distribution for the transformation . Let's define , and thus we have that . Now, since g is a monotonous function, we can use the following rule to find the distribution of :

It's clear that . Therefore, using the above rule, we get that the pdf for Y is:

After some simplification, we get that:

So that's the pdf for the Inverted Gamma distribution.

b. In order to be a pdf, it must integrate to 1. Let's calculate the ...

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