ANL 252 SUSS Python for Data Analytics Worksheet

ANL252Python for Data Analytics
Tutor-Marked Assignment
July 2021 Presentation
ANL252
Tutor-Marked Assignment
TUTOR-MARKED ASSIGNMENT (TMA)
This assignment is worth 18% of the final mark for ANL252 Python for Data Analytics.
The cut-off date for this assignment is 8 August 2021, 2355hrs.
Up to 25 marks of penalties will be imposed for inappropriate or poor paraphrasing. For
serious cases, they will be investigated by the examination department. More information on
effective
paraphrasing
strategies
can
be
found
on
https://academicguides.waldenu.edu/writingcenter/evidence/paraphrase/effective.
Note to Students:
You are to include the following particulars in your submission: Course Code, Title of the
TMA, SUSS PI No., Your Name, and Submission Date.
Question
The probability density function (pdf) of the normal distribution is given by
𝑓𝑓𝑋𝑋 (𝑥𝑥) =
1
√2𝜋𝜋𝜎𝜎 2
exp �−
(𝑥𝑥 − 𝜇𝜇)2
�,
2𝜎𝜎 2
where x is the observed value of a random variable X, π is the value of the constant pi (≈
3.14159), µ is the mean and σ2 is the variance of the distribution, and exp() is the exponential
function.
As an analyst and Python programming specialist, you are asked to construct a program to
compute the probability of a normally distributed random variable given the distribution
mean and variance.
Note: Include your Python program code in the answers and show them in the “Consolas” or
“Courier New” fonts (size 12). Make a screenshot of the program output if required.
(a)
Prepare the programming of the function by importing the “math” package into your
program which is needed for the square root and the exponential functions. You can
also find a much more exact value for π in the math package. Refer to the official
website of the math package https://docs.python.org/3/library/math.html for more
details of the functions.
(1 mark)
(b)
Employ a Python program to ask the user to enter the mean and variance of the
distribution. Tell the user that the mean can be any value between minus infinity (–∞)
and plus infinity (+∞), but the variance must be a value larger than 0. Integrate certain
control mechanism to ensure that the variance condition is fulfilled, and the input is
numeric. If the user press ENTER without providing any values, the program will
automatically set µ to 0 and σ2 to 1.
(15 marks)
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)
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ANL252
Tutor-Marked Assignment
(c)
Design an input screen for the user to enter the value of X. Tell the user that the value
can be any value between minus infinity (–∞) and plus infinity (+∞). Integrate certain
control mechanism to ensure that the input is numeric.
(4 marks)
(d)
Construct a user-defined function using the formula of the probability density function
given above to compute the corresponding probability density fX(x) based on the user
inputs in (b) and (c).
(15 marks)
(e)
Use formatted printing to display the result of (d) to the user.
(f)
The probability of a normally distributed random variable X smaller equal to a value k
can be determined by the cumulative distribution function (cdf) which can be
approximated by the weighted sum of the density function of X from –∞ to k:
(10 marks)
𝑃𝑃(𝑋𝑋 ≤ 𝑘𝑘) ≈ 𝛼𝛼[𝑓𝑓𝑋𝑋 (𝑎𝑎) + ⋯ + 𝑓𝑓𝑋𝑋 (𝑘𝑘 − 2𝛼𝛼) + 𝑓𝑓𝑋𝑋 (𝑘𝑘 − 𝛼𝛼) + 𝑓𝑓𝑋𝑋 (𝑘𝑘)],
where α is the range of each step and a is a number close to –∞. For example, if we
choose α = 0.01, a = -100 and k = 0, then the summation is
𝑃𝑃(𝑋𝑋 ≤ 0) ≈ 0.01[𝑓𝑓𝑋𝑋 (−100) + ⋯ + 𝑓𝑓𝑋𝑋 (−0.02) + 𝑓𝑓𝑋𝑋 (−0.01) + 𝑓𝑓𝑋𝑋 (0)]
The accuracy of the approximation increases with the decrease of the step range α. That
means, the smaller the step width is, the more terms we can put between a and k, and
the more accurate is the probability.
Design a program to compute P(X ≤ k) where k is the value that the user entered in (c).
Use the user input in (b) as the values for the parameters µ and σ2 and display the result
to the user by formatted printing. Furthermore, check the results for k = 0, 1.64, and
1.96 with a mean of 0 and variance of 1 as well.
(25 marks)
(g)
Explain your program for the calculation of P(X ≤ k) in (f) in less than 200 words.
(15 marks)
(h)
Create a dictionary to store the probabilities of the normal distribution for x = {-5, -4.9,
-4.8, …, 0, 0.1, 0.2, …, 4.8, 4.9, 5} where x are the keys, and the corresponding
probabilities are the values of the dictionary. The distribution mean here is 0 and
variance is 1. Print all the probabilities (with the corresponding x) of those x’s between
-2 and 2 with a step width of 0.5 from the dictionary onto the screen.
(15 marks)
—- END OF ASSIGNMENT —-
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS)
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