Lab4_Random_Variables-Solution

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Lab4_Random_Variables-Solution October 5, 2021 0.1 Lab #4 Random Variables Full Name: replace text here SID: replace text here NOTE: Replace the “…” with your code/answer and make sure insert code in cells where indicated. Save the file as a PDF file and name it “Lab4_FirstnameLastname”. Upload and submit your report via Gradescope by Thursday 7:00PM . Make sure to run all cells so that the output is visible for every cell before turning in the notebook. Please remember to label all axes with the quantity and units being plotted. ACADEMIC INTEGRITY Berkeley Campus Code of Student Conduct (http://sa.berkeley.edu/student-code-of-conduct): “The Chancellor may impose discipline for the commission or attempted commission (including aiding or abetting in the commission or attempted commission) of the following types of violations by students, as well as such other violations as may be specified in campus regulations: 102.01 Academic Dishonesty: All forms of academic misconduct including but not limited to cheating, fabrication, plagiarism, or facilitating academic dishonesty.” [1]: # import packages import matplotlib.pyplot as plt import numpy as np import pandas as pd from scipy import integrate 0.2 Problem 1 Discrete Random Variable The probability mass function (PMF) of a random variable is given in the first row of the following table. 1a) Calculate p x (3) . Print an array of the PMF values including p x (3) . You can print outputs by using the print( ) function. [2]: # Calculate px(3) PMF_X_3 = 1-0.28-0.24-0.01-0.2-0.005-0.008 print (PMF_X_3) 0.25699999999999995 1

[3]: # Print an array of all PMF values PMF_X = np . array([ 0.28 , 0.24 , 0.01 , PMF_X_3, 0.2 , 0.005 , 0.008 ]) print (PMF_X) [0.28 0.24 0.01 0.257 0.2 0.005 0.008] 1b) Compute the values of the cumulative distribution function (CDF) in the cell below and print them. You can print outputs by using the print( ) function. [4]: # Print an array of CDF CDF_X = np . cumsum(PMF_X) print (CDF_X) [0.28 0.52 0.53 0.787 0.987 0.992 1. ] 1c) Plot the PMF and the CDF of using ‘bar’ and step’ commands from matplotlib.pyplot, respec- tively. Use figsize=[12,6]. Use the subplot function to create a 1x2 figure (meaning two seperate graphs will print out). Make sure you create two separate plots for PMF and CDF. [5]: # insert your code for making the plots here X = np . arange( 7 ) plt . figure(figsize = ( 12 , 6 )) plt . subplot( 121 ) plt . bar(X, PMF_X) plt . title( 'PMF' ) plt . xlabel( 'X' ) plt . ylabel( 'Probability' ) plt . subplot( 122 ) plt . step(X, CDF_X) plt . title( 'CDF' ) plt . xlabel( 'X' ) plt . ylabel( 'CDF' ) plt . show() 2

1d) Use the function from the lab lecture, compute the mean, the standard deviation, and the skewness coeffcient of the discrete random variable X in the cell below and print each value. Remember to use the specific corresponding equations from lecture. Do not use built-in functions from python or any library Hint: Variables you have defined before may be useful here. [6]: # Calculate the statistics in 1d) here mean_X = np . dot(X, PMF_X) std_X = np . dot((X - mean_X) **2 , PMF_X) **0.5 skew_X = np . dot(((X - mean_X) / std_X) **3 , PMF_X) print ( 'The mean of X: ' , mean_X) print ( 'The standard deviation of X: ' , std_X) print ( 'The skewness coefficient of X: ' , skew_X) The mean of X: 1.904 The standard deviation of X: 1.6064818704236907 The skewness coefficient of X: 0.18527586393793305 0.3 Problem 2 Continuous Random Variables Consider the probability density function (PDF) of a continuous random variable: In this part, you will need to define and call functions in python, as we’ve learned in the demo code. 2a) Compute the value of c . 3

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[7]: c = 1/ ( 1.5* ( 1- np . exp( -14/1.5 ))) print (c) 0.6667256232057891 2b) Code the above PDF as a python function that takes y as input and returns f Y ( y ) . Consider which parameter(s) you want to use when defining this python function. You only need to care about conditions where y ∈ [0 , 14] in this question. [8]: # Define the PDF here def PDF_Y (y): return 0.6667* np . exp( - y /1.5 ) 2c) Derive the CDF of Y and write a python function for this CDF. You only need to care about conditions where y ∈ [0 , 14] in this question. [9]: # Define the CDF here def CDF_Y (y): return 1.5*0.6667* ( 1- np . exp( - y /1.5 )) 2d) Plot the PDF and the CDF for y ∈ [0 , 14] using the functions you define in part b) and part c). Use the subplot function to create a 1x2 figure (meaning two seperate graphs will print out).Set the figsize=[12, 6]. Make sure you create two separate plots for PDF and CDF. [10]: # Plot the PDF and CDF here Y = np . arange( 0 , 14 , 0.1 ) plt . figure(figsize = ( 12 , 6 )) plt . subplot( 121 ) plt . plot(Y, PDF_Y(Y)) plt . title( 'PDF' ) plt . xlabel( 'Y' ) plt . ylabel( 'Probability' ) plt . subplot( 122 ) plt . plot(Y, CDF_Y(Y)) plt . title( 'CDF' ) plt . xlabel( 'Y' ) plt . ylabel( 'CDF' ) [10]: Text(0, 0.5, 'CDF') 4

2e) What is the probability of y between 3 and 5? Use the proper function you defined in previous parts to calculate. Then print your result. Hint: Use the integrate.quad() function. Details can be found in the demo code for this lab. [11]: # The probability of y between 3 and 5 P_Y_3_5 = integrate . quad(PDF_Y, 3 , 5 )[ 0 ] print (P_Y_3_5) 0.09966627295385477 2f) Compute the mean, the standard deviation, and the skewness coeffcient of this continuous random variable using the equations provided in the lecture. Do not use the built-in functions from python or any library. Hint: Define three functions for the three statistics. Use the proper parameters to define the functions. Since this is a continuous variable, you can use integrate.quad() method to calculate the statistic. Details can be found in the demo code for this lab. [12]: # Define function for the mean def func_mean (y): return y * PDF_Y(y) # Define function for the std def func_std (y, mean): return (y - mean) **2* PDF_Y(y) # Define function for the skewness def func_skew (y, mean, std): 5

return ((y - mean) / std) **3* PDF_Y(y) [13]: # Use the functions you defined above and the integrate.quad() function # to get the statistics for this continuous variable mean_Y = integrate . quad(func_mean, 0 , 14 )[ 0 ] print ( 'The mean of Y: ' , mean_Y) The mean of Y: 1.4987043131448554 [14]: std_Y = (integrate . quad(func_std, 0 , 14 , args = (mean_Y,))[ 0 ]) **0.5 print ( 'The standard deviation of Y: ' , std_Y) The standard deviation of Y: 1.494181864259477 [15]: skew_Y = integrate . quad(func_skew, 0 , 14 , args = (mean_Y, std_Y,))[ 0 ] print ( 'The skewness coefficient of Y: ' , skew_Y) The skewness coefficient of Y: 1.9507316690314322 0.4 SUMITE YOUR WORK (DOWNLOAD AS PDF) 1. Make sure to run all cells in the notebook after you complete. You can do this by going to Cell > Run All. This makes sure that all your visuals and answers show up in the report you submit. 2. In jupyter notebook, go to “File > Download as > PDF via LaTex(.pdf)” to generate a PDF file. Name the PDF file as “Lab2_FirstnameLastname”. This should work if you are using the link we provided for the assignment. 3. If you are not using the link we provided for the assignment and have trouble generating the PDF file from your local jupyter notebook, • Option 1: use datahub.berkeley.edu . Log in with your CalNet credentials. Upload the ipynb file with your outputs and results to jupyterhub. Then follow above instruction (step 2). • Option 2: go to “File > Download as > Markdown(.md)” to generate a markdown file. Then, go to https://typora.io/ and download the Typora (markdown editor). Open the markdown file with Typora and go to “File > Export > PDF” to generate a PDF file (step 2). 5. Name it “Lab4_FirstnameLastname”. Upload and submit the PDF file via Gradescope. [ ]: 6

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