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Three simple computation experiments are outlined below. You can use any language that you are
comfortable with. The key objective of this experiment is to make you think about many of the questions
that are given at the end. If you have done some of this formally through other means, this will be good
revision. If you are seeing these questions for the first time, then do spend time thinking about them as
a means of knowing the motivations for many of the concepts that we develop in this course. We will
formally address them as we proceed through the course. Of course, increasing degrees of complexity will
be developed. Finally, real life data is not as cleanly available as the examples suggest. Some examples
will be introduced as we progress through the course.
1. There are 10,000 students in a IITB. A cellphone company is considering designing some deals for
the students and wants to determine the average call-minute consumption per day per student.
Collecting the information from every student is costly, time consuming, and can be prone to errors.
It may not even be possible. The obvious ‘short cut’ is to obtain the data from a small number of
students, take the sample mean, and declare that that is the true value of the data that is being
sought. However, this is clearly a guess. For the guess to be good, we need to answer at least two
questions. Assume that you have a budget to collect data from K students.
• How to select the K students to collect their data?
• How does the guess ‘improve’ as a function of K? And hence, what is a good K?
Here are some options for choosing the K students from whom to collect the data.
(a) Ask the first K students that you can find as soon as you enter the campus.
(b) Choose an arbitrary point in the campus and ask K students from there. Here arbitrary means
that you can pick any point that you like.
(c) Randomly select K from the 10,000 people in the colony. You can visualise a random selection
to be the result of the following experiment. Put all the 10,000 names in a pot, mix the pot
thoroughly and pick a name. Repeat K times. You can use a random number generator to
pick the name.
File hw1a.txt contains the ‘actual’ weekly call-minutes from the 10,0000 users. Assume that this
data is not available to you. For a given K, write a program to simulate the three scenarios described
above.
(a) For each of the three scenarios, write a program to obtain the samples from this list, and
calculate the sample average. The program should repeat this experiment fifty times and make
a scatter-plot of the twenty points for each of the three scenarios, i.e., mark a dot for every one
of the fifty points on an a suitable scale.
(b) Repeat the experiment for K = 10, 20, 50, 100, 200. (There is one scatter plot for every combination of K and the method of selecting the K. Now answer the following questions.
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i. What is your guess for the actual average and the actual standard deviation? And what
is the actual value?
ii. Each of the fifty repetitions can be seen to be a separate survey. If you could do the survey
only once for a given K,
A. Which of the above three schemes would you use in practice to determined the best
guess?
B. If you were allowed to choose the value of K, what value would you choose? And
how sure would you be of the actual values of the average? What kind of quantitative
measure would you use to describe your “sureness of the estimate from the single survey
of K samples?”
2. Assume that Kohli always calls Heads when he goes out for the toss and he has to call. What
do you think he will assume the probability of the coin toss coming up heads to be? Assume
that he believes that his opponent does not have the powers or Shakuni or is using the coin
that Amitabh Bachchan used in Sholay (if you do not know what that is, see around 3:50 and
13:05 https://www.youtube.com/watch?v=QmLiyVT lLc. Now imagine that he sees the sequence
in hw1b1.txt that contains the result of 100 tosses of a coin.
(a) At what point in the sequence does he begin to doubt his initial assumption?
(b) And when can he be sure that his initial assumption is wrong, if it is indeed wrong?
(c) Repeat for three more coins. The results of 100 tosses of these coins are in files hw1b2.txt–
hw1b4.txt.
(d) At every point in the sequence, when you are determining whether you are sure of your initial
assumption or not, think about how you would describe the “degree of sureness,” i.e., provide
a quantitative description to your ‘hunch.’
3. Given N pairs of data (xi
, yi) for i = 1, N you have done straight line fitting on this data of the form
y = ax+b. Recall that you determine a and b to minimise root mean square error PN
i=1(yi−axi−b)
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.
Consider the data given in hw1c1.txt. Here xi represents the height of person i in centimeters of
IITB student i and yi
is the weight of the person in kgs. Fifty IIT students were sampled and their
height and weight recorded in the file. Write a program to fit the straight line to this data, i.e.,
determine a and b. The data in hw1c1.txt is used construct a model. Now let us see how to use
this data.
(a) Assume now that you have determined the model described above. You see a person of height
x. You can use the model to predict this person’s height. Let your prediction be denoted by ˆy.
When you measure the actual weight, now denoted by y what can you say about the difference
between the true height y and your intelligent guess ˆy using model.
(b) Now assume that you meet many people of height x and for each of these you guess their weight
y, ˆ measure the true value y and obtain the error y − y. ˆ What kind of properties would you like
for these errors. Specifically, comment on the average and the standard deviation of the errors.
You can use the data in file hw1c2.txt that contains 25 values of y for three different values
of x to help arrive at your answer.
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