E3. To be completed after lesson 13

Exercise 3.1

What is the relationship between a cumul ative distribution function (CDF) and a probability density function (PDF)? What is the relationship between as CDF and a probability mass function (PMF)?

Exercise 3.2

Say I am interested in modeling the time it takes for a certain process to happen in a cell. For reasons that you will understand in future lessons, we might model the time \(t\) to be Exponentially distributed, having probability density function

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

  1. Write down the CDF, \(F(t)\).

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

Exercise 3.3

How would you expect each of the following to be distributed?

a) The amount of time between repressor-operator binding events.

b) The number of times a repressor binds its operator in a given hour.

c) The amount of time (in total minutes of baseball played) between no-hitters in Major League Baseball.

d) The number of no-hitters in a Major League Baseball season.

e) The winning times of the Belmont Stakes.

To answer this question, try to match these stories to the stories of named distributions. For those of you not familiar with baseball, a no-hitter is a game in which a team concedes no hits to the opposing team. There have only been a few hundred no-hitters in over 200,000 MLB games. The Belmont Stakes is a major horse race that has been run each year for over 150 years.

Exercise 3.4

Say I have three distributions:

  • Exponential, β=1

  • Normal, μ=1, σ=1

  • Cauchy, µ=1, σ=1

Say I draw numbers out of each of these distributions. Rank order the distributions, lowest to highest, in terms of how likely I am do draw a number greater than 10. You do not need to calculate anything to answer this question.

Exercise 3.5

Now draw a million numbers out of each of the three distributions in Exercise 3.4. Which, if any, had numbers greater than ten? Was your intuition from exercise 3.4 correct?

Exercise 3.6

Write down any questions or points of confusion that you have.