Homework 3.2. The Exponential distribution (10 pts)

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You may submit this problem as a PDF, either with clearly typed mathematics or with neat handwriting, if you like.

Say I am interested in modeling the time it between repressor-operator binding events in a cell. We might model the time \(t\) to be Exponentially distributed, having probability density function

\begin{align} f(t;\beta) = \beta \mathrm{e}^{-\beta t}. \end{align}

a) Why might we model \(t\) as being Exponentially distributed?

b) Write down the CDF, \(F(t;\beta)\).

c) Say you are interested instead in the reciprocal time, \(k = 1/t\). Write down the PDF for \(k\), \(f(k;\beta)\).