Midterm Exam

FIZ371 - Scientific & Technical Calculations | 07/12/2022

Emre S. Tasci emre.tasci@hacettepe.edu.tr Eng. Physics Dept.
Hacettepe University
Ankara, Turkey

Midterm Exam#

You can solve the problems analytically or stochastically, it’s up to you. You can present your solution having been calculated on a paper, or as a jupyter notebook, or as a mixture of both.

  • Please prepare one jupyter notebook per question.

  • You can use any predefined function from numpy, scipy, math, pandas, random, collections and matplotlib modules – for any other external module/library you should first ask for permission.

  • Before submitting, download your notebooks as html as well as ipynb format and zip them all.

  • Name your zip file as <YourName>_FIZ137_MT.zip (e.g., EmreTasci_FIZ137_MT.zip)

1. Probabilities#

Please choose one from the following two questions:

1A - Dice Rolling#

Suppose I have a box with a 6-sided die, an 8-sided die, and a 12-sided die. I choose one of the dice at random, roll it four times, and get 1, 3, 5, and 7. What is the probability that I chose the 8-sided die?

1B - Coin Tossing#

Upon tossing a coin 8 times, calculate the probability that the number of heads will be greater than the number of tails.

2. Distributions#

Please choose one from the following two questions:

2A - Borel Distribution#

The Borel distribution is a discrete probability distribution defined as:

\[P_{mu}(n) = \frac{e^{-\mu n}(\mu n)^{n-1}}{n!}\]

for its variable \(n=1,2,3,\dots\), and its parameter \(\mu\in[0,1]\)

Show that:
a) it’s normalized
b) its mean is equal to (/converges to): \(\frac{1}{1-\mu}\)
c) its variance is equal to: \(\frac{\mu}{(1-\mu)^3}\)

Bonus: Plot its graph for \(\mu_1 = 0.2\) and \(\mu_2 = 0.7\)

2B - Gaussian Distribution#

Sample \(N=1000\) data points (\(\{x_i\}\) from a Gaussian distribution with \(\mu = 72\) and \(\sigma=15\). Then, add a random noise of 10% to each data point and fit it to a Gaussian (by deriving the most suitable \(\mu'\) and \(\sigma'\), calculate the error \(S_r'\) defined as:

\[S_r' = \sum_{i}{\left(y_i - G(x_i;\mu',\sigma')\right)^2}\]

and compare it to the error calculated with respect to your original \((\mu,\sigma)\):

\[S_r = \sum_{i}{\left(y_i - G(x_i;\mu,\sigma)\right)^2}\]

3. Game Theory#

Please choose one from the following two questions:

3A - Path to Success#

Tomorrow, early in the morning, you are to take an important exam. But the love of your life has asked for a meeting that night, which will definitely and negatively effect your exam performance on the following day. While you are pondering on this situation, unknown to you, your significant other receives a phone call for a meeting early in the morning from a company she/he had applied. And then, all the means of communications between you two disappears all of a sudden.

Construct the pay-off table stating your reasons. Draw the reaction-correspondence graph, indicate Nash equilibrium points if there are any. Calculate the probabilities that each of you will attend tonight’s meeting or not (if any of the probabilities is 0 or 1 then re-construct the pay-off table, and redo the calculations!)

3B - Nash in Real Life#

 

\(B_1\)

\(B_2\)

\(B_3\)

\(A_1\)

20,20

30,30

10,20

\(A_2\)

40,30

50,20

20,25

\(A_3\)

30,20

60,0

30,10

a) Find the Nash equilibrium point(s), if any.
b) Treat this as the preference table for a Gale-Shapley matching (e.g., \(A_2\) prefers \(B_2\) with a 50 score but \(B_2\) prefers \(A_2\) with a 20 score) and find a stable matching, show that it’s stable.

Attention: the higher the score, the higher the preference! It’s a score, not a rank!

4. “You vs. You”#

Propose and solve a question from any topic you wish as long as it isn’t one of the topics of the above questions (and given that we have covered it in our lectures 8).
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