Probability theory and random variables

Author

Swethaa

Published

November 14, 2023

Probability Distribution

Content

What is Probability Distribution?

The result of a random variable is unknown in probability distribution. Here, the observation of the result is referred to as realization. It's a function that converts Sample Space into State Space, a real number space. They may be Continuous or Discrete.In essence, a probability distribution is the collection of all potential results from a random experiment or occurrence.

Random Variables

The sample space of the random experiment serves as the domain for a random variable, which is a real-valued function. As X(sample space) = Real number, it is expressed. The idea of random variables is important to understand because, occasionally, we are not just interested in the likelihood of an event happening, but also in the total number of events connected to the random experiment.The following example will help you better understand the significance of random variables:

Why do we need Random Variables?


So now we flip our coin 3 times, and we want to answer some questions.

  1. What is the probability of getting exactly 3 heads?
  2. What is the probability of getting less than 3 heads?
  3. What is the probability of getting more than 1 head?

Then our general way of writing would be:

  1. P(Probability of getting exactly 3 heads when we flip a coin 3 times)
  2. P(Probability of getting less than 3 heads when we flip a coin 3 times)
  3. P(Probability of getting more than 1 head when we flip a coin 3 times)

Definition of Random Variable

A random variable is a real valued function whose domain is the sample space of a random experiment

Types of Random Variables in Probability Distribution

There are following two types of
  • Discrete Random Variables
  • Continuous Random Variables

Discrete Random Variables in Probability Distribution

A discrete random variable is a kind of random variable in probability theory that has a countable or finite set of possible values. In addition to being represented by whole numbers or integers, these values can also be represented by other discrete values.

A discrete random variable would be the number of heads that result from tossing a coin three times. This variable can have one of three possible values: 0, 1, 2, or 3.

there are various other examples of random discrete variables.
  • The number of cars that pass through a given intersection in an hour.
  • The number of defective items in a shipment of goods.
  • The number of red balls drawn in a sample of 10 balls taken from a jar containing both red and blue balls.

Different Types of Probability Distributions

  • Discrete Probability Distributions for discrete variables
  • Cumulative Probability Distribution for continuous variables
import warnings
import seaborn as sns
import matplotlib.pyplot as plt
warnings.filterwarnings('ignore')
import numpy as np
import pandas as pd
from IPython.display import Image
from sklearn.preprocessing import StandardScaler
from sklearn.cluster import DBSCAN
sns.set(style="darkgrid", palette="pastel", color_codes=True)
sns.set_context('talk')

The dataset, 'Walmart_Store_sales.csv', is loaded into a Pandas DataFrame, providing a structured and manipulable format for the data.The first five rows of the dataset are displayed, offering an initial glimpse into the data's structure and contents.

df = pd.read_csv('Walmart_Store_sales.csv')
df.head(5)
Store Date Weekly_Sales Holiday_Flag Temperature Fuel_Price CPI Unemployment
0 1 05-02-2010 1643690.90 0 42.31 2.572 211.096358 8.106
1 1 12-02-2010 1641957.44 1 38.51 2.548 211.242170 8.106
2 1 19-02-2010 1611968.17 0 39.93 2.514 211.289143 8.106
3 1 26-02-2010 1409727.59 0 46.63 2.561 211.319643 8.106
4 1 05-03-2010 1554806.68 0 46.50 2.625 211.350143 8.106

This provides an overview of the column names.

df.columns
Index(['Store', 'Date', 'Weekly_Sales', 'Holiday_Flag', 'Temperature',
       'Fuel_Price', 'CPI', 'Unemployment'],
      dtype='object')

This code command provides a concise summary of a DataFrame in Python, displaying information such as the number of non-null values, data types, and memory usage.

df.info()
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 6435 entries, 0 to 6434
Data columns (total 8 columns):
 #   Column        Non-Null Count  Dtype  
---  ------        --------------  -----  
 0   Store         6435 non-null   int64  
 1   Date          6435 non-null   object 
 2   Weekly_Sales  6435 non-null   float64
 3   Holiday_Flag  6435 non-null   int64  
 4   Temperature   6435 non-null   float64
 5   Fuel_Price    6435 non-null   float64
 6   CPI           6435 non-null   float64
 7   Unemployment  6435 non-null   float64
dtypes: float64(5), int64(2), object(1)
memory usage: 402.3+ KB

This code retrieves the column names of a DataFrame in pandas, providing a quick overview of the dataset's features

This code provides a quick overview of the central tendency, dispersion, and distribution of the numerical data in the DataFrame.

df.describe()
Store Weekly_Sales Holiday_Flag Temperature Fuel_Price CPI Unemployment
count 6435.000000 6.435000e+03 6435.000000 6435.000000 6435.000000 6435.000000 6435.000000
mean 23.000000 1.046965e+06 0.069930 60.663782 3.358607 171.578394 7.999151
std 12.988182 5.643666e+05 0.255049 18.444933 0.459020 39.356712 1.875885
min 1.000000 2.099862e+05 0.000000 -2.060000 2.472000 126.064000 3.879000
25% 12.000000 5.533501e+05 0.000000 47.460000 2.933000 131.735000 6.891000
50% 23.000000 9.607460e+05 0.000000 62.670000 3.445000 182.616521 7.874000
75% 34.000000 1.420159e+06 0.000000 74.940000 3.735000 212.743293 8.622000
max 45.000000 3.818686e+06 1.000000 100.140000 4.468000 227.232807 14.313000

df.shape It returns a tuple representing the shape of the DataFrame in the form (number of rows, number of columns)

df.shape
(6435, 8)

df.isnull() is a useful tool for data cleaning and exploration to identify and handle missing values in a dataset. 

df.isnull()
Store Date Weekly_Sales Holiday_Flag Temperature Fuel_Price CPI Unemployment
0 False False False False False False False False
1 False False False False False False False False
2 False False False False False False False False
3 False False False False False False False False
4 False False False False False False False False
... ... ... ... ... ... ... ... ...
6430 False False False False False False False False
6431 False False False False False False False False
6432 False False False False False False False False
6433 False False False False False False False False
6434 False False False False False False False False

6435 rows × 8 columns

dropna is used to dropping rows with missing values from the DataFrame. .sum() checks and counts the number of missing values in each column of the DataFrame 

df.dropna(inplace =True)
df.isnull().sum()
Store           0
Date            0
Weekly_Sales    0
Holiday_Flag    0
Temperature     0
Fuel_Price      0
CPI             0
Unemployment    0
dtype: int64

To Check if there are any duplicated rows in the DataFrame. 

df.duplicated().any()
False

This code provides a visual representation of the distribution of the variable Weekly_Sales along with its fitted normal distribution. 

from scipy.stats import norm
variable_of_interest = 'Weekly_Sales'
plt.figure(figsize=(8, 6))
sns.histplot(df[variable_of_interest], kde=True, bins=30, color='blue', stat='density')
mu, std = df[variable_of_interest].mean(), df[variable_of_interest].std()
xmin, xmax = plt.xlim()
x = np.linspace(xmin, xmax, 100).round(2)
p = norm.pdf(x, mu, std).round(2)
plt.plot(x, p, 'k', linewidth=2)
print(variable_of_interest,x,p)
plt.xlabel(variable_of_interest)
plt.ylabel('Probability Density')
plt.title(f'Probability Density Function for {variable_of_interest}')
plt.show()
Weekly_Sales [  29551.24   69647.91  109744.58  149841.25  189937.92  230034.58
  270131.25  310227.92  350324.59  390421.26  430517.93  470614.6
  510711.27  550807.94  590904.6   631001.27  671097.94  711194.61
  751291.28  791387.95  831484.62  871581.29  911677.96  951774.62
  991871.29 1031967.96 1072064.63 1112161.3  1152257.97 1192354.64
 1232451.31 1272547.98 1312644.64 1352741.31 1392837.98 1432934.65
 1473031.32 1513127.99 1553224.66 1593321.33 1633418.   1673514.66
 1713611.33 1753708.   1793804.67 1833901.34 1873998.01 1914094.68
 1954191.35 1994288.02 2034384.68 2074481.35 2114578.02 2154674.69
 2194771.36 2234868.03 2274964.7  2315061.37 2355158.04 2395254.7
 2435351.37 2475448.04 2515544.71 2555641.38 2595738.05 2635834.72
 2675931.39 2716028.06 2756124.72 2796221.39 2836318.06 2876414.73
 2916511.4  2956608.07 2996704.74 3036801.41 3076898.08 3116994.74
 3157091.41 3197188.08 3237284.75 3277381.42 3317478.09 3357574.76
 3397671.43 3437768.1  3477864.76 3517961.43 3558058.1  3598154.77
 3638251.44 3678348.11 3718444.78 3758541.45 3798638.12 3838734.78
 3878831.45 3918928.12 3959024.79 3999121.46] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
 0. 0. 0. 0.]

This code efficiently organizes subplots for each column, creates histograms with KDE, overlays PDFs based on normal distributions, and displays the visualizations for multiple columns in your DataFrame. Adjustments are made for cases where the number of columns is odd. 

sns.set(style="whitegrid")
df.drop('Date', axis=1, inplace=True)
num_columns = len(df.columns)
num_rows = (num_columns // 2) + (num_columns % 2)  
num_cols = 2
fig, axes = plt.subplots(num_rows, num_cols, figsize=(8, 3 * num_rows))
if num_rows == 1:
    axes = axes.reshape(1, -1)
for i, column in enumerate(df.columns):
    row_idx = i // num_cols
    col_idx = i % num_cols
    sns.histplot(df[column], kde=True, bins=30, color='blue', stat='density', ax=axes[row_idx, col_idx])
    mu, std = df[column].mean(), df[column].std()
    xmin, xmax = axes[row_idx, col_idx].get_xlim()
    x = np.linspace(xmin, xmax, 100)
    p = norm.pdf(x, mu, std)
    axes[row_idx, col_idx].plot(x, p, 'k', linewidth=2)
    print(column )
    print(x,p)
    axes[row_idx, col_idx].set_xlabel(column)
    axes[row_idx, col_idx].set_ylabel('Probability Density')
    axes[row_idx, col_idx].set_title(f'PDF for {column}')

if num_columns % 2 == 1:
    fig.delaxes(axes[-1, -1])
plt.tight_layout()
plt.show()
Store
[-1.2        -0.71111111 -0.22222222  0.26666667  0.75555556  1.24444444
  1.73333333  2.22222222  2.71111111  3.2         3.68888889  4.17777778
  4.66666667  5.15555556  5.64444444  6.13333333  6.62222222  7.11111111
  7.6         8.08888889  8.57777778  9.06666667  9.55555556 10.04444444
 10.53333333 11.02222222 11.51111111 12.         12.48888889 12.97777778
 13.46666667 13.95555556 14.44444444 14.93333333 15.42222222 15.91111111
 16.4        16.88888889 17.37777778 17.86666667 18.35555556 18.84444444
 19.33333333 19.82222222 20.31111111 20.8        21.28888889 21.77777778
 22.26666667 22.75555556 23.24444444 23.73333333 24.22222222 24.71111111
 25.2        25.68888889 26.17777778 26.66666667 27.15555556 27.64444444
 28.13333333 28.62222222 29.11111111 29.6        30.08888889 30.57777778
 31.06666667 31.55555556 32.04444444 32.53333333 33.02222222 33.51111111
 34.         34.48888889 34.97777778 35.46666667 35.95555556 36.44444444
 36.93333333 37.42222222 37.91111111 38.4        38.88888889 39.37777778
 39.86666667 40.35555556 40.84444444 41.33333333 41.82222222 42.31111111
 42.8        43.28888889 43.77777778 44.26666667 44.75555556 45.24444444
 45.73333333 46.22222222 46.71111111 47.2       ] [0.00541385 0.00580306 0.00621145 0.00663916 0.00708628 0.00755281
 0.00803865 0.00854362 0.00906746 0.0096098  0.01017016 0.01074795
 0.01134248 0.01195295 0.01257845 0.01321793 0.01387027 0.01453418
 0.01520832 0.01589119 0.01658121 0.0172767  0.01797588 0.01867687
 0.01937772 0.0200764  0.02077083 0.02145885 0.02213827 0.02280686
 0.02346238 0.02410257 0.02472517 0.02532794 0.02590868 0.0264652
 0.0269954  0.02749724 0.02796875 0.02840806 0.02881343 0.0291832
 0.02951587 0.02981006 0.03006456 0.0302783  0.03045039 0.03058009
 0.03066687 0.03071035 0.03071035 0.03066687 0.03058009 0.03045039
 0.0302783  0.03006456 0.02981006 0.02951587 0.0291832  0.02881343
 0.02840806 0.02796875 0.02749724 0.0269954  0.0264652  0.02590868
 0.02532794 0.02472517 0.02410257 0.02346238 0.02280686 0.02213827
 0.02145885 0.02077083 0.0200764  0.01937772 0.01867687 0.01797588
 0.0172767  0.01658121 0.01589119 0.01520832 0.01453418 0.01387027
 0.01321793 0.01257845 0.01195295 0.01134248 0.01074795 0.01017016
 0.0096098  0.00906746 0.00854362 0.00803865 0.00755281 0.00708628
 0.00663916 0.00621145 0.00580306 0.00541385]
Weekly_Sales
[  29551.24         69647.90888889  109744.57777778  149841.24666667
  189937.91555556  230034.58444444  270131.25333333  310227.92222222
  350324.59111111  390421.26        430517.92888889  470614.59777778
  510711.26666667  550807.93555556  590904.60444444  631001.27333333
  671097.94222222  711194.61111111  751291.28        791387.94888889
  831484.61777778  871581.28666667  911677.95555556  951774.62444444
  991871.29333333 1031967.96222222 1072064.63111111 1112161.3
 1152257.96888889 1192354.63777778 1232451.30666667 1272547.97555556
 1312644.64444444 1352741.31333333 1392837.98222222 1432934.65111111
 1473031.32       1513127.98888889 1553224.65777778 1593321.32666667
 1633417.99555556 1673514.66444444 1713611.33333333 1753708.00222222
 1793804.67111111 1833901.34       1873998.00888889 1914094.67777778
 1954191.34666667 1994288.01555556 2034384.68444444 2074481.35333333
 2114578.02222222 2154674.69111111 2194771.36       2234868.02888889
 2274964.69777778 2315061.36666667 2355158.03555556 2395254.70444444
 2435351.37333333 2475448.04222222 2515544.71111111 2555641.38
 2595738.04888889 2635834.71777778 2675931.38666667 2716028.05555556
 2756124.72444444 2796221.39333333 2836318.06222222 2876414.73111111
 2916511.4        2956608.06888889 2996704.73777778 3036801.40666667
 3076898.07555556 3116994.74444444 3157091.41333333 3197188.08222222
 3237284.75111111 3277381.42       3317478.08888889 3357574.75777778
 3397671.42666667 3437768.09555556 3477864.76444444 3517961.43333333
 3558058.10222222 3598154.77111111 3638251.44       3678348.10888889
 3718444.77777778 3758541.44666667 3798638.11555556 3838734.78444444
 3878831.45333333 3918928.12222222 3959024.79111111 3999121.46      ] [1.39199576e-07 1.57821620e-07 1.78033976e-07 1.99823745e-07
 2.23151138e-07 2.47947041e-07 2.74111062e-07 3.01510196e-07
 3.29978205e-07 3.59315802e-07 3.89291739e-07 4.19644822e-07
 4.50086885e-07 4.80306715e-07 5.09974861e-07 5.38749261e-07
 5.66281556e-07 5.92223934e-07 6.16236337e-07 6.37993805e-07
 6.57193757e-07 6.73562975e-07 6.86864061e-07 6.96901164e-07
 7.03524775e-07 7.06635436e-07 7.06186218e-07 7.02183909e-07
 6.94688834e-07 6.83813346e-07 6.69719026e-07 6.52612685e-07
 6.32741318e-07 6.10386174e-07 5.85856147e-07 5.59480692e-07
 5.31602514e-07 5.02570228e-07 4.72731236e-07 4.42424991e-07
 4.11976849e-07 3.81692637e-07 3.51854057e-07 3.22715000e-07
 2.94498815e-07 2.67396526e-07 2.41565985e-07 2.17131888e-07
 1.94186595e-07 1.72791628e-07 1.52979752e-07 1.34757523e-07
 1.18108158e-07 1.02994630e-07 8.93628642e-08 7.71449294e-08
 6.62621467e-08 5.66280265e-08 4.81509835e-08 4.07367808e-08
 3.42906771e-08 2.87192586e-08 2.39319561e-08 1.98422529e-08
 1.63686015e-08 1.34350713e-08 1.09717572e-08 8.91497668e-09
 7.20729076e-09 5.79737888e-09 4.63979866e-09 3.69465929e-09
 2.92723417e-09 2.30753526e-09 1.80986856e-09 1.41238643e-09
 1.09664950e-09 8.47207833e-10 6.51208245e-10 4.98032436e-10
 3.78968556e-10 2.86917167e-10 2.16131317e-10 1.61989430e-10
 1.20799035e-10 8.96288985e-11 6.61668508e-11 4.86004936e-11
 3.55180165e-11 2.58264396e-11 1.86847884e-11 1.34499180e-11
 9.63294103e-12 6.86445362e-12 4.86699453e-12 3.43339326e-12
 2.40987243e-12 1.68295446e-12 1.16938761e-12 8.08448608e-13]
Holiday_Flag
[-0.05       -0.03888889 -0.02777778 -0.01666667 -0.00555556  0.00555556
  0.01666667  0.02777778  0.03888889  0.05        0.06111111  0.07222222
  0.08333333  0.09444444  0.10555556  0.11666667  0.12777778  0.13888889
  0.15        0.16111111  0.17222222  0.18333333  0.19444444  0.20555556
  0.21666667  0.22777778  0.23888889  0.25        0.26111111  0.27222222
  0.28333333  0.29444444  0.30555556  0.31666667  0.32777778  0.33888889
  0.35        0.36111111  0.37222222  0.38333333  0.39444444  0.40555556
  0.41666667  0.42777778  0.43888889  0.45        0.46111111  0.47222222
  0.48333333  0.49444444  0.50555556  0.51666667  0.52777778  0.53888889
  0.55        0.56111111  0.57222222  0.58333333  0.59444444  0.60555556
  0.61666667  0.62777778  0.63888889  0.65        0.66111111  0.67222222
  0.68333333  0.69444444  0.70555556  0.71666667  0.72777778  0.73888889
  0.75        0.76111111  0.77222222  0.78333333  0.79444444  0.80555556
  0.81666667  0.82777778  0.83888889  0.85        0.86111111  0.87222222
  0.88333333  0.89444444  0.90555556  0.91666667  0.92777778  0.93888889
  0.95        0.96111111  0.97222222  0.98333333  0.99444444  1.00555556
  1.01666667  1.02777778  1.03888889  1.05      ] [1.40046748e+00 1.42809627e+00 1.45350893e+00 1.47656879e+00
 1.49715040e+00 1.51514060e+00 1.53043964e+00 1.54296203e+00
 1.55263737e+00 1.55941100e+00 1.56324451e+00 1.56411614e+00
 1.56202091e+00 1.55697075e+00 1.54899432e+00 1.53813679e+00
 1.52445938e+00 1.50803880e+00 1.48896655e+00 1.46734800e+00
 1.44330154e+00 1.41695737e+00 1.38845644e+00 1.35794911e+00
 1.32559390e+00 1.29155605e+00 1.25600620e+00 1.21911892e+00
 1.18107131e+00 1.14204163e+00 1.10220788e+00 1.06174653e+00
 1.02083124e+00 9.79631669e-01 9.38312373e-01 8.97031788e-01
 8.55941308e-01 8.15184478e-01 7.74896287e-01 7.35202577e-01
 6.96219566e-01 6.58053481e-01 6.20800303e-01 5.84545622e-01
 5.49364595e-01 5.15322001e-01 4.82472389e-01 4.50860313e-01
 4.20520637e-01 3.91478921e-01 3.63751853e-01 3.37347742e-01
 3.12267053e-01 2.88502967e-01 2.66041977e-01 2.44864492e-01
 2.24945454e-01 2.06254955e-01 1.88758851e-01 1.72419357e-01
 1.57195634e-01 1.43044348e-01 1.29920202e-01 1.17776443e-01
 1.06565332e-01 9.62385821e-02 8.67477564e-02 7.80446342e-02
 7.00815358e-02 6.28116113e-02 5.61890929e-02 5.01695113e-02
 4.47098778e-02 3.97688340e-02 3.53067705e-02 3.12859175e-02
 2.76704077e-02 2.44263168e-02 2.15216802e-02 1.89264920e-02
 1.66126853e-02 1.45540987e-02 1.27264289e-02 1.11071736e-02
 9.67556475e-03 8.41249563e-03 7.30044179e-03 6.32337876e-03
 5.46669711e-03 4.71711638e-03 4.06259877e-03 3.49226376e-03
 2.99630421e-03 2.56590478e-03 2.19316309e-03 1.87101420e-03
 1.59315856e-03 1.35399382e-03 1.14855058e-03 9.72432206e-04]
Temperature
[ -7.17        -6.03444444  -4.89888889  -3.76333333  -2.62777778
  -1.49222222  -0.35666667   0.77888889   1.91444444   3.05
   4.18555556   5.32111111   6.45666667   7.59222222   8.72777778
   9.86333333  10.99888889  12.13444444  13.27        14.40555556
  15.54111111  16.67666667  17.81222222  18.94777778  20.08333333
  21.21888889  22.35444444  23.49        24.62555556  25.76111111
  26.89666667  28.03222222  29.16777778  30.30333333  31.43888889
  32.57444444  33.71        34.84555556  35.98111111  37.11666667
  38.25222222  39.38777778  40.52333333  41.65888889  42.79444444
  43.93        45.06555556  46.20111111  47.33666667  48.47222222
  49.60777778  50.74333333  51.87888889  53.01444444  54.15
  55.28555556  56.42111111  57.55666667  58.69222222  59.82777778
  60.96333333  62.09888889  63.23444444  64.37        65.50555556
  66.64111111  67.77666667  68.91222222  70.04777778  71.18333333
  72.31888889  73.45444444  74.59        75.72555556  76.86111111
  77.99666667  79.13222222  80.26777778  81.40333333  82.53888889
  83.67444444  84.81        85.94555556  87.08111111  88.21666667
  89.35222222  90.48777778  91.62333333  92.75888889  93.89444444
  95.03        96.16555556  97.30111111  98.43666667  99.57222222
 100.70777778 101.84333333 102.97888889 104.11444444 105.25      ] [2.50099169e-05 3.13053732e-05 3.90372716e-05 4.84946649e-05
 6.00153584e-05 7.39920024e-05 9.08784879e-05 1.11196557e-04
 1.35542509e-04 1.64593863e-04 1.99115786e-04 2.39967087e-04
 2.88105535e-04 3.44592214e-04 4.10594620e-04 4.87388169e-04
 5.76355772e-04 6.78985103e-04 7.96863217e-04 9.31668152e-04
 1.08515720e-03 1.25915152e-03 1.45551696e-03 1.67614073e-03
 1.92290410e-03 2.19765091e-03 2.50215220e-03 2.83806719e-03
 3.20690105e-03 3.60995993e-03 4.04830419e-03 4.52270052e-03
 5.03357400e-03 5.58096142e-03 6.16446680e-03 6.78322075e-03
 7.43584484e-03 8.12042243e-03 8.83447733e-03 9.57496145e-03
 1.03382527e-02 1.11201639e-02 1.19159636e-02 1.27204092e-02
 1.35277924e-02 1.43319972e-02 1.51265691e-02 1.59047956e-02
 1.66597961e-02 1.73846202e-02 1.80723517e-02 1.87162167e-02
 1.93096942e-02 1.98466250e-02 2.03213177e-02 2.07286492e-02
 2.10641566e-02 2.13241182e-02 2.15056228e-02 2.16066234e-02
 2.16259759e-02 2.15634608e-02 2.14197869e-02 2.11965786e-02
 2.08963445e-02 2.05224312e-02 2.00789608e-02 1.95707555e-02
 1.90032502e-02 1.83823960e-02 1.77145564e-02 1.70063997e-02
 1.62647884e-02 1.54966701e-02 1.47089711e-02 1.39084949e-02
 1.31018286e-02 1.22952574e-02 1.14946902e-02 1.07055961e-02
 9.93295275e-03 9.18120791e-03 8.45425241e-03 7.75540590e-03
 7.08741381e-03 6.45245499e-03 5.85215894e-03 5.28763144e-03
 4.75948733e-03 4.26788893e-03 3.81258889e-03 3.39297596e-03
 3.00812253e-03 2.65683264e-03 2.33768940e-03 2.04910094e-03
 1.78934397e-03 1.55660438e-03 1.34901445e-03 1.16468608e-03]
Fuel_Price
[2.3722     2.39437778 2.41655556 2.43873333 2.46091111 2.48308889
 2.50526667 2.52744444 2.54962222 2.5718     2.59397778 2.61615556
 2.63833333 2.66051111 2.68268889 2.70486667 2.72704444 2.74922222
 2.7714     2.79357778 2.81575556 2.83793333 2.86011111 2.88228889
 2.90446667 2.92664444 2.94882222 2.971      2.99317778 3.01535556
 3.03753333 3.05971111 3.08188889 3.10406667 3.12624444 3.14842222
 3.1706     3.19277778 3.21495556 3.23713333 3.25931111 3.28148889
 3.30366667 3.32584444 3.34802222 3.3702     3.39237778 3.41455556
 3.43673333 3.45891111 3.48108889 3.50326667 3.52544444 3.54762222
 3.5698     3.59197778 3.61415556 3.63633333 3.65851111 3.68068889
 3.70286667 3.72504444 3.74722222 3.7694     3.79157778 3.81375556
 3.83593333 3.85811111 3.88028889 3.90246667 3.92464444 3.94682222
 3.969      3.99117778 4.01335556 4.03553333 4.05771111 4.07988889
 4.10206667 4.12424444 4.14642222 4.1686     4.19077778 4.21295556
 4.23513333 4.25731111 4.27948889 4.30166667 4.32384444 4.34602222
 4.3682     4.39037778 4.41255556 4.43473333 4.45691111 4.47908889
 4.50126667 4.52344444 4.54562222 4.5678    ] [0.08635809 0.09569467 0.10579341 0.11668518 0.12839821 0.14095758
 0.15438463 0.16869643 0.18390516 0.20001756 0.21703437 0.23494981
 0.25375105 0.27341781 0.2939219  0.3152269  0.33728792 0.36005139
 0.38345498 0.40742761 0.43188958 0.45675275 0.48192096 0.5072904
 0.53275024 0.55818332 0.58346693 0.60847371 0.63307269 0.65713036
 0.68051182 0.70308203 0.72470709 0.74525553 0.76459965 0.78261681
 0.79919072 0.81421272 0.82758292 0.83921133 0.84901889 0.8569383
 0.86291484 0.866907   0.86888694 0.86884083 0.86676897 0.86268586
 0.85661996 0.8486134  0.83872149 0.82701206 0.81356471 0.7984699
 0.78182794 0.76374786 0.74434627 0.72374606 0.70207514 0.67946512
 0.65604998 0.63196477 0.60734436 0.58232217 0.55702903 0.53159211
 0.50613388 0.48077124 0.45561471 0.43076775 0.40632619 0.38237777
 0.35900181 0.336269   0.31424127 0.29297178 0.27250504 0.25287709
 0.23411574 0.21624095 0.19926519 0.18319395 0.1680262  0.15375494
 0.14036775 0.12784736 0.11617225 0.10531718 0.09525378 0.0859511
 0.07737609 0.06949417 0.06226961 0.05566601 0.04964668 0.04417499
 0.03921471 0.03473023 0.03068687 0.02705102]
CPI
[121.00555966 122.12965751 123.25375537 124.37785322 125.50195107
 126.62604893 127.75014678 128.87424463 129.99834249 131.12244034
 132.24653819 133.37063605 134.4947339  135.61883175 136.74292961
 137.86702746 138.99112531 140.11522317 141.23932102 142.36341887
 143.48751673 144.61161458 145.73571243 146.85981029 147.98390814
 149.10800599 150.23210385 151.3562017  152.48029955 153.60439741
 154.72849526 155.85259311 156.97669097 158.10078882 159.22488667
 160.34898453 161.47308238 162.59718023 163.72127809 164.84537594
 165.96947379 167.09357165 168.2176695  169.34176735 170.46586521
 171.58996306 172.71406091 173.83815877 174.96225662 176.08635447
 177.21045233 178.33455018 179.45864803 180.58274589 181.70684374
 182.83094159 183.95503945 185.0791373  186.20323515 187.32733301
 188.45143086 189.57552871 190.69962657 191.82372442 192.94782227
 194.07192013 195.19601798 196.32011583 197.44421369 198.56831154
 199.69240939 200.81650725 201.9406051  203.06470295 204.18880081
 205.31289866 206.43699651 207.56109437 208.68519222 209.80929007
 210.93338793 212.05748578 213.18158363 214.30568149 215.42977934
 216.55387719 217.67797505 218.8020729  219.92617075 221.05026861
 222.17436646 223.29846431 224.42256217 225.54666002 226.67075787
 227.79485573 228.91895358 230.04305143 231.16714929 232.29124714] [0.00443956 0.00460365 0.00476991 0.00493814 0.00510814 0.00527968
 0.00545253 0.00562645 0.00580119 0.00597647 0.00615203 0.00632758
 0.00650283 0.00667749 0.00685124 0.00702379 0.00719481 0.00736399
 0.00753099 0.0076955  0.0078572  0.00801575 0.00817083 0.00832212
 0.0084693  0.00861206 0.00875007 0.00888306 0.00901071 0.00913274
 0.00924887 0.00935884 0.0094624  0.00955931 0.00964933 0.00973225
 0.00980789 0.00987605 0.00993658 0.00998932 0.01003416 0.01007098
 0.01009969 0.01012022 0.01013253 0.01013657 0.01013236 0.01011988
 0.01009918 0.0100703  0.01003331 0.00998831 0.00993541 0.00987472
 0.0098064  0.00973062 0.00964754 0.00955738 0.00946034 0.00935664
 0.00924654 0.00913028 0.00900813 0.00888037 0.00874728 0.00860916
 0.00846632 0.00831905 0.00816768 0.00801252 0.0078539  0.00769215
 0.00752758 0.00736052 0.00719131 0.00702025 0.00684768 0.0066739
 0.00649923 0.00632397 0.00614841 0.00597286 0.00579758 0.00562286
 0.00544896 0.00527614 0.00510462 0.00493466 0.00476646 0.00460025
 0.00443621 0.00427453 0.00411538 0.00395893 0.00380532 0.00365469
 0.00350716 0.00336284 0.00322183 0.00308421]
Unemployment
[ 3.3573      3.47323333  3.58916667  3.7051      3.82103333  3.93696667
  4.0529      4.16883333  4.28476667  4.4007      4.51663333  4.63256667
  4.7485      4.86443333  4.98036667  5.0963      5.21223333  5.32816667
  5.4441      5.56003333  5.67596667  5.7919      5.90783333  6.02376667
  6.1397      6.25563333  6.37156667  6.4875      6.60343333  6.71936667
  6.8353      6.95123333  7.06716667  7.1831      7.29903333  7.41496667
  7.5309      7.64683333  7.76276667  7.8787      7.99463333  8.11056667
  8.2265      8.34243333  8.45836667  8.5743      8.69023333  8.80616667
  8.9221      9.03803333  9.15396667  9.2699      9.38583333  9.50176667
  9.6177      9.73363333  9.84956667  9.9655     10.08143333 10.19736667
 10.3133     10.42923333 10.54516667 10.6611     10.77703333 10.89296667
 11.0089     11.12483333 11.24076667 11.3567     11.47263333 11.58856667
 11.7045     11.82043333 11.93636667 12.0523     12.16823333 12.28416667
 12.4001     12.51603333 12.63196667 12.7479     12.86383333 12.97976667
 13.0957     13.21163333 13.32756667 13.4435     13.55943333 13.67536667
 13.7913     13.90723333 14.02316667 14.1391     14.25503333 14.37096667
 14.4869     14.60283333 14.71876667 14.8347    ] [0.0099562  0.01157924 0.01341552 0.01548376 0.01780273 0.02039097
 0.02326646 0.02644625 0.02994602 0.03377966 0.03795882 0.04249241
 0.04738613 0.05264199 0.05825787 0.06422707 0.07053795 0.0771736
 0.08411161 0.09132388 0.09877658 0.10643018 0.11423965 0.12215469
 0.13012018 0.13807669 0.14596116 0.15370765 0.16124819 0.16851379
 0.17543542 0.18194508 0.18797695 0.19346841 0.19836122 0.20260246
 0.2061455  0.20895089 0.21098705 0.2122309  0.21266824 0.21229409
 0.2111127  0.20913757 0.2063911  0.20290423 0.19871583 0.19387197
 0.18842513 0.18243318 0.17595843 0.16906649 0.16182522 0.15430362
 0.14657072 0.1386946  0.1307414  0.12277442 0.11485341 0.10703384
 0.0993664  0.09189655 0.08466426 0.07770379 0.07104369 0.06470682
 0.05871051 0.05306679 0.04778274 0.04286082 0.03829932 0.03409282
 0.03023264 0.02670732 0.02350314 0.02060453 0.01799453 0.01565524
 0.01356814 0.01171445 0.01007546 0.00863274 0.00736841 0.00626528
 0.00530699 0.00447814 0.00376433 0.00315224 0.00262961 0.00218527
 0.00180909 0.00149196 0.00122573 0.00100317 0.00081789 0.00066428
 0.00053747 0.00043321 0.00034784 0.00027823]

from scipy.stats import rv_histogram
column_of_interest = 'Weekly_Sales'
hist, bin_edges = np.histogram(df[column_of_interest], bins=30, density=True)
pmf = rv_histogram((hist, bin_edges))
x_values = np.linspace(df[column_of_interest].min(), df[column_of_interest].max(), 100).round(2)
pmf_values = pmf.pdf(x_values)
print("X values\n",x_values)
print("pmf values \n",pmf_values)
plt.bar(bin_edges[:-1], hist, width=np.diff(bin_edges), alpha=0.7, label='Histogram')
plt.plot(x_values, pmf_values, color='red', label='PMF', linewidth=2)
plt.title(f'Probability Mass Function (PMF) for {column_of_interest}')
plt.xlabel(column_of_interest)
plt.ylabel('Probability')
plt.legend()
plt.show()
X values
 [ 209986.25  246437.77  282889.28  319340.8   355792.32  392243.84
  428695.35  465146.87  501598.39  538049.9   574501.42  610952.94
  647404.46  683855.97  720307.49  756759.01  793210.52  829662.04
  866113.56  902565.08  939016.59  975468.11 1011919.63 1048371.14
 1084822.66 1121274.18 1157725.7  1194177.21 1230628.73 1267080.25
 1303531.77 1339983.28 1376434.8  1412886.32 1449337.83 1485789.35
 1522240.87 1558692.39 1595143.9  1631595.42 1668046.94 1704498.45
 1740949.97 1777401.49 1813853.01 1850304.52 1886756.04 1923207.56
 1959659.07 1996110.59 2032562.11 2069013.63 2105465.14 2141916.66
 2178368.18 2214819.69 2251271.21 2287722.73 2324174.25 2360625.76
 2397077.28 2433528.8  2469980.31 2506431.83 2542883.35 2579334.87
 2615786.38 2652237.9  2688689.42 2725140.93 2761592.45 2798043.97
 2834495.49 2870947.   2907398.52 2943850.04 2980301.56 3016753.07
 3053204.59 3089656.11 3126107.62 3162559.14 3199010.66 3235462.18
 3271913.69 3308365.21 3344816.73 3381268.24 3417719.76 3454171.28
 3490622.8  3527074.31 3563525.83 3599977.35 3636428.86 3672880.38
 3709331.9  3745783.42 3782234.93 3818686.45]
pmf values 
 [5.42589256e-07 5.42589256e-07 5.42589256e-07 5.42589256e-07
 6.96322879e-07 6.96322879e-07 6.96322879e-07 9.53406836e-07
 9.53406836e-07 9.53406836e-07 6.75652812e-07 6.75652812e-07
 6.75652812e-07 6.75652812e-07 4.80579056e-07 4.80579056e-07
 4.80579056e-07 6.09766974e-07 6.09766974e-07 6.09766974e-07
 6.71777175e-07 6.71777175e-07 6.71777175e-07 6.71777175e-07
 4.75411539e-07 4.75411539e-07 4.75411539e-07 4.93497847e-07
 4.93497847e-07 4.93497847e-07 5.92972544e-07 5.92972544e-07
 5.92972544e-07 4.43114559e-07 4.43114559e-07 4.43114559e-07
 4.43114559e-07 3.19094158e-07 3.19094158e-07 3.19094158e-07
 2.31246374e-07 2.31246374e-07 2.31246374e-07 3.11342883e-07
 3.11342883e-07 3.11342883e-07 3.11342883e-07 3.20386037e-07
 3.20386037e-07 3.20386037e-07 2.41581407e-07 2.41581407e-07
 2.41581407e-07 1.27896039e-07 1.27896039e-07 1.27896039e-07
 1.27896039e-07 3.61726171e-08 3.61726171e-08 3.61726171e-08
 2.32538253e-08 2.32538253e-08 2.32538253e-08 1.55025502e-08
 1.55025502e-08 1.55025502e-08 1.55025502e-08 1.03350335e-08
 1.03350335e-08 1.03350335e-08 1.67944294e-08 1.67944294e-08
 1.67944294e-08 6.45939591e-09 6.45939591e-09 6.45939591e-09
 2.58375836e-09 2.58375836e-09 2.58375836e-09 2.58375836e-09
 0.00000000e+00 0.00000000e+00 0.00000000e+00 1.29187918e-09
 1.29187918e-09 1.29187918e-09 2.58375836e-09 2.58375836e-09
 2.58375836e-09 2.58375836e-09 5.16751673e-09 5.16751673e-09
 5.16751673e-09 2.58375836e-09 2.58375836e-09 2.58375836e-09
 3.87563755e-09 3.87563755e-09 3.87563755e-09 0.00000000e+00]

from scipy.stats import geom
selected_column = 'Temperature'
temperature_threshold = 60
success_data = df[df[selected_column] > temperature_threshold]
p = len(success_data) / len(df)
print('probability of success ',p)
k_values = np.arange(1, 11)
pmf_values = geom.pmf(k_values, p)
plt.bar(k_values, pmf_values, align='center', alpha=0.7)
plt.title(f'Geometric Distribution PMF for {selected_column}')
plt.xlabel('Number of Trials Until First Success (k)')
plt.ylabel('Probability')
plt.show()
probability of success  0.5435897435897435

import statsmodels.api as sm
columns = df.columns
fig, axes = plt.subplots(nrows=len(columns), ncols=1, figsize=(8, 4 * len(columns)))
for i, column in enumerate(columns):
    ecdf = sm.distributions.ECDF(df[column])
    x = sorted(df[column].unique())
    y = ecdf(x)
    print(column)
    print(x,y)
    axes[i].step(x, y, label=column)
    axes[i].set_title(f'Cumulative Distribution Function of {column}')
    axes[i].set_xlabel(column)
    axes[i].set_ylabel('Cumulative Probability')
    axes[i].legend()
plt.tight_layout()
plt.show()
Store
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] [0.02222222 0.04444444 0.06666667 0.08888889 0.11111111 0.13333333
 0.15555556 0.17777778 0.2        0.22222222 0.24444444 0.26666667
 0.28888889 0.31111111 0.33333333 0.35555556 0.37777778 0.4
 0.42222222 0.44444444 0.46666667 0.48888889 0.51111111 0.53333333
 0.55555556 0.57777778 0.6        0.62222222 0.64444444 0.66666667
 0.68888889 0.71111111 0.73333333 0.75555556 0.77777778 0.8
 0.82222222 0.84444444 0.86666667 0.88888889 0.91111111 0.93333333
 0.95555556 0.97777778 1.        ]
Weekly_Sales
[209986.25, 213538.32, 215359.21, 219804.85, 220060.35, 224031.19, 224294.39, 224639.76, 224806.96, 226702.36, 229731.98, 230519.49, 231155.9, 231319.96, 231976.84, 232769.09, 233543.08, 234175.92, 234218.03, 234793.12, 236157.12, 236920.49, 237095.82, 237129.81, 237405.82, 238084.08, 238172.66, 238433.53, 238875.26, 238915.05, 239198.36, 239206.26, 239431.85, 240044.57, 241937.11, 242047.03, 242456.39, 242526.7, 242771.37, 242813.51, 242901.21, 243477.03, 243948.82, 244338.31, 244856.44, 244899.2, 245435.8, 246124.61, 246277.18, 246970.97, 247234.47, 247672.56, 248051.53, 248561.86, 248603.3, 249134.32, 249246.8, 249798.75, 251294.5, 251327.67, 251732.92, 252349.6, 252709.58, 252734.31, 253050.1, 253205.89, 253731.13, 254412.34, 255996.47, 256091.32, 256235.19, 257031.19, 257361.3, 258427.39, 258533.12, 259419.91, 259527.75, 259638.35, 260636.71, 261131.09, 261666.29, 261837.2, 261851.74, 262159.13, 262407.57, 262717.71, 262789.95, 262893.76, 263263.02, 263917.85, 264214.12, 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Fuel_Price
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 9.97668998e-01 9.98135198e-01 9.99067599e-01 1.00000000e+00]
CPI
[126.064, 126.0766452, 126.0854516, 126.0892903, 126.1019355, 126.1069032, 126.1119032, 126.114, 126.1145806, 126.1266, 126.1283548, 126.1360645, 126.1392, 126.1454667, 126.1498065, 126.1518, 126.1602258, 126.1843871, 126.1900333, 126.2085484, 126.2346, 126.2791667, 126.2898, 126.3266774, 126.3805667, 126.3815484, 126.4364194, 126.4420645, 126.4713333, 126.4912903, 126.4962581, 126.5262857, 126.5461613, 126.5522857, 126.5621, 126.5782857, 126.6019032, 126.6034839, 126.6042857, 126.6050645, 126.6066452, 126.6072, 126.6692667, 126.7313333, 126.7934, 126.8794839, 126.9835806, 127.0876774, 127.1917742, 127.3009355, 127.4404839, 127.5800323, 127.7195806, 127.859129, 127.99525, 128.13, 128.26475, 128.3995, 128.5121935, 128.6160645, 128.7199355, 128.8238065, 128.9107333, 128.9553, 128.9998667, 129.0357097, 129.0432, 129.0444333, 129.0490323, 129.0623548, 129.0663, 129.0756774, 129.089, 129.0894, 129.1125, 129.1338387, 129.1507742, 129.1677097, 129.1846452, 129.2015806, 129.2405806, 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Unemployment
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 3.25407925e-01 3.26651127e-01 3.31002331e-01 3.35042735e-01
 3.39083139e-01 3.40326340e-01 3.42346542e-01 3.50116550e-01
 3.54156954e-01 3.56177156e-01 3.58197358e-01 3.60683761e-01
 3.62703963e-01 3.66744367e-01 3.68764569e-01 3.70784771e-01
 3.78865579e-01 3.80885781e-01 3.82750583e-01 3.87101787e-01
 3.89121989e-01 3.91142191e-01 3.93317793e-01 3.97358197e-01
 3.98601399e-01 3.99222999e-01 4.03263403e-01 4.06993007e-01
 4.08236208e-01 4.12587413e-01 4.16938617e-01 4.20979021e-01
 4.22999223e-01 4.25019425e-01 4.27195027e-01 4.31235431e-01
 4.33255633e-01 4.35275835e-01 4.37296037e-01 4.43356643e-01
 4.45376845e-01 4.47241647e-01 4.49261849e-01 4.51437451e-01
 4.55167055e-01 4.57187257e-01 4.59362859e-01 4.61383061e-01
 4.65734266e-01 4.67754468e-01 4.69774670e-01 4.71639472e-01
 4.73815074e-01 4.75990676e-01 4.84693085e-01 4.88733489e-01
 4.94794095e-01 4.98834499e-01 5.02874903e-01 5.04895105e-01
 5.06915307e-01 5.08780109e-01 5.16860917e-01 5.20901321e-01
 5.22921523e-01 5.25097125e-01 5.27117327e-01 5.29137529e-01
 5.30380730e-01 5.31002331e-01 5.33022533e-01 5.35042735e-01
 5.37218337e-01 5.39393939e-01 5.46853147e-01 5.48873349e-01
 5.53224553e-01 5.55244755e-01 5.59285159e-01 5.71406371e-01
 5.72649573e-01 5.74669775e-01 5.76689977e-01 5.77933178e-01
 5.80108780e-01 5.84149184e-01 5.92851593e-01 5.99378399e-01
 6.03418803e-01 6.05905206e-01 6.09945610e-01 6.18026418e-01
 6.20046620e-01 6.22066822e-01 6.23310023e-01 6.25330225e-01
 6.25951826e-01 6.29836830e-01 6.31857032e-01 6.35897436e-01
 6.40248640e-01 6.42424242e-01 6.43667444e-01 6.49728050e-01
 6.51748252e-01 6.52991453e-01 6.57964258e-01 6.59207459e-01
 6.61227661e-01 6.63714064e-01 6.65889666e-01 6.71950272e-01
 6.77544678e-01 6.81585082e-01 6.83605284e-01 6.85780886e-01
 6.87801088e-01 6.89821290e-01 6.91686092e-01 6.97746698e-01
 6.99766900e-01 7.06293706e-01 7.07536908e-01 7.11577312e-01
 7.15617716e-01 7.19658120e-01 7.23698524e-01 7.27738928e-01
 7.29759130e-01 7.35353535e-01 7.39083139e-01 7.43123543e-01
 7.45299145e-01 7.49339549e-01 7.53690754e-01 7.54933955e-01
 7.59285159e-01 7.61305361e-01 7.61926962e-01 7.63170163e-01
 7.69386169e-01 7.71406371e-01 7.72027972e-01 7.76379176e-01
 7.80419580e-01 7.86169386e-01 7.88189588e-01 7.90365190e-01
 7.94094794e-01 7.97358197e-01 7.99378399e-01 8.01398601e-01
 8.03574204e-01 8.07614608e-01 8.09479409e-01 8.11499611e-01
 8.15540016e-01 8.17560218e-01 8.19580420e-01 8.21600622e-01
 8.24087024e-01 8.28438228e-01 8.30924631e-01 8.34965035e-01
 8.36985237e-01 8.38850039e-01 8.43201243e-01 8.45221445e-01
 8.45843046e-01 8.49883450e-01 8.51126651e-01 8.52369852e-01
 8.54545455e-01 8.56565657e-01 8.60606061e-01 8.62781663e-01
 8.64801865e-01 8.66822067e-01 8.68842269e-01 8.70862471e-01
 8.73348873e-01 8.77389277e-01 8.81429681e-01 8.85470085e-01
 8.89510490e-01 8.91996892e-01 8.96037296e-01 8.98057498e-01
 9.00233100e-01 9.02253302e-01 9.03496503e-01 9.04739705e-01
 9.08780109e-01 9.10800311e-01 9.12665113e-01 9.17016317e-01
 9.18881119e-01 9.22610723e-01 9.24630925e-01 9.26806527e-01
 9.30846931e-01 9.35198135e-01 9.41258741e-01 9.47319347e-01
 9.53379953e-01 9.59440559e-01 9.65967366e-01 9.72027972e-01
 9.75757576e-01 9.81351981e-01 9.87412587e-01 9.93473193e-01
 1.00000000e+00]

from scipy.stats import hypergeom
population_size = len(df)
sample_size = 10
num_columns = len(df.columns)
num_rows = (num_columns // 2) + (num_columns % 2)
num_cols = 2

fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 3 * num_rows))
if num_rows == 1:
    axes = axes.reshape(1, -1)
for i, column in enumerate(df.columns):
    row_idx = i // num_cols
    col_idx = i % num_cols

    numeric_data = pd.to_numeric(df[column], errors='coerce')
    numeric_data = numeric_data.dropna()
    successes_in_population = np.sum(numeric_data > 0)
    print(column)
    print('successes',successes_in_population)
    k_values = np.arange(0, min(sample_size, successes_in_population) + 1)

    pmf_values = hypergeom.pmf(k_values, population_size, successes_in_population, sample_size)
    print(f'Hypergeometric Distribution PMF for {column}',pmf_values)
    axes[row_idx, col_idx].bar(k_values, pmf_values, align='center', alpha=0.7)
    axes[row_idx, col_idx].set_title(f'Hypergeometric Distribution PMF for {column}')
    axes[row_idx, col_idx].set_xlabel('Number of Successes in Draws (k)')
    axes[row_idx, col_idx].set_ylabel('Probability')

if num_columns % 2 == 1:
    fig.delaxes(axes[-1, -1])

plt.tight_layout()
plt.show()
Store
successes 6435
Hypergeometric Distribution PMF for Store [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.]
Weekly_Sales
successes 6435
Hypergeometric Distribution PMF for Weekly_Sales [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.]
Holiday_Flag
successes 450
Hypergeometric Distribution PMF for Holiday_Flag [4.84091496e-01 3.64526729e-01 1.23226745e-01 2.46261099e-02
 3.22190575e-03 2.88355177e-04 1.78785674e-05 7.58282840e-07
 2.10546109e-08 3.45593360e-10 2.54647739e-12]
Temperature
successes 6434
Hypergeometric Distribution PMF for Temperature [0.       0.       0.       0.       0.       0.       0.       0.
 0.       0.001554 0.998446]
Fuel_Price
successes 6435
Hypergeometric Distribution PMF for Fuel_Price [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.]
CPI
successes 6435
Hypergeometric Distribution PMF for CPI [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.]
Unemployment
successes 6435
Hypergeometric Distribution PMF for Unemployment [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.]

from scipy.stats import norm
numeric_columns = df.select_dtypes(include=['int64', 'float64']).columns

fig, axes = plt.subplots(nrows=len(numeric_columns), figsize=(8, 4 * len(numeric_columns)))
fig.subplots_adjust(hspace=0.5)

for i, column in enumerate(numeric_columns):
    sns.histplot(df[column], kde=True, ax=axes[i])
    axes[i].set_title(f'Distribution of {column}')
    axes[i].set_xlabel(column)
    axes[i].set_ylabel('Density')
    mu, std = norm.fit(df[column])
    xmin, xmax = plt.xlim()
    x = np.linspace(xmin, xmax, 100)
    p = norm.pdf(x, mu, std)
    axes[i].plot(x, p, 'k', linewidth=2)
    axes[i].legend(['Gaussian fit'])

plt.tight_layout()
plt.show()