Homework 4.3: Change of parametrization and change of variables (10 pts)

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a) The Negative Binomial distribution notoriously have many parametrizations. Systems biologists have a preferred parametrization when describing bursty gene expression (see Cai, et al., 2006). Bursty gene expression can be observed when a gene is turned “on,” e.g. by repressor unbinding, and many RNA transcripts are made. It is then turned “off” by repressor binding, and no transcripts are made for a while.

Referring to the parametrization in the Distribution Explorer systems biologists use the parameter \(\alpha\), which is the burst frequency. They prefer not to use the parameter \(\beta\), but instead use the parameter \(b = 1/\beta\), which is the burst size, the mean number of transcripts per burst.

Write down the probability mass function for the Negative Binomial distribution parametrized by \(\alpha\) and \(b\).

b) The Beta distribution describes variables on the interval [0, 1]. Say a variable \(\theta\) is Beta distributed. Write down the PDF for \(x = 1/\theta\).