Practice Problems

Lecture 15

Answer each number in a separate cell

Rename this notebook with your last name and the lecture

ex. CychB_16

Turn-in this notebook on TritonEd by the end of class

1. Binomial functions

  • Assume that the probability of getting a winning lottery ticket ($p$) is 1 in 20 and you have money to purchase up to 10 tickets ($n$).
    • Calculate the probability of purchasing one winning lottery ticket.
    • Create a list of probabilites. The probability of purchasing 1 winning ticket, 2 winning tickets, 3 winning tickets, ... up to 10 winning tickets.
    • Plot the probability against the number of winning tickets purchased as a green bar plot. Label both axes

2. Monte Carlo simulations with random.binomial( )

  • Again, assume that the probability of getting a winning lottery ticket ($p$) is 1 in 20 and you have money to purchase up to 10 tickets ($n$).
    • Run 100 simulations ($nmc$) of the scenario
    • Plot the simulated results as a histogram and the theoretical distribution as a line graph
    • Add a title to the plot
    • Add a legend
    • Add a label to the x-axis and y-axis

3. Uniform distributions

  • Calculate the theoretical distribution of getting a particular azimuth between 0 and 360 when measuring, for example, the direction of a strike - assume that each result is equally likely (a uniform distribution between 0 and 360)
  • Perform a Monte Carlo simulation with $n=30$ trials.
  • Plot your theoretical and simulated results as a bar and histogram plot respectively.
  • Try this again using the random.seed() function.

4. Normal distributions

  • Calculate the theoretical distribution of grades on an exam with a mean of 50% and a standard deviation of $\pm$ 20.
  • Simulate the results of an exam taken by 35 students
  • Calculate the mean and standard deviation of your simulated results.
  • Plot theoretical and simulated results as a bar graph and histogram respectively.
  • Plot a solid line representing the cutoffs for As, Bs, Cs, Ds, and Fs
  • Add a title, x-label, y-label, and legend

5. Log-normal distributions

  • Simulate a grain size distribution that is drawn from a log normal distribution with 1000 grains and a mean and standard deviation of 10 and 0.1 microns respectively.
  • Plot the distribution as a histogram (density set to True).
  • Calculate the mean, standard deviation, expectation and variance of the distribution.