Random Variables

A Random variable is a variable that takes on numerical values as a result of random expriments of mesuarement, associates a numerical value with each possible outcom .. R.V's must have numberical values.

  • Discrete Random Variable: has a finite number of values or an infinate sequence of values(0,2,3,...) AND the differnces between coutcomes are menaningful
    • Die throw can have 1,2,3,4,6 and each is a meaningfully different.
  • Continuous Random Variable: ahs a nearly infinite numbers of outcomes that cannot be easiily counted AND the differences between teh outcomes are NOT meaningful
    • With average income, the difference between $40,00 and $40,001 is not meaningful

Discrete Probability Distribution

The probability distribution for a random variable X describes how assigned to each outcome for the random variable. Let 0=Heads and 1=Tails for a coin flip. so ouir discrete random variable x is described as: x = 0,1

  • The Probability for each outcome is described by a discrete porbability funcion denoted by P(x) $$\sum P(x) = 1$$ sum of all RV Probabilities P(x) must equal 1

Uniform Probability Distribution 

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.stats import uniform

#Generating Uniform random numbers
from scipy.stats import uniform

# We can generate random variables/numbers
# from uniform distribution from uniform distribution’s rvs function like uniform.rvs.
# Here we are generating 1000 Random Variables (rvs) between 0 and 10.
data_uniform = uniform.rvs(size=1000, loc=0, scale=10)   

len(data_uniform)

1000
data_uniform[:20]

array([7.33092523, 6.91163852, 6.81076766, 6.10710074, 6.3161025 ,
       8.99572593, 5.78687982, 2.91428895, 5.54023576, 7.51997417,
       7.85882577, 1.15105694, 6.13031743, 1.06657784, 1.61417764,
       1.58547971, 6.65943099, 3.0200822 , 9.14553805, 7.58889625])
ax = sns.distplot(data_uniform,
                  bins=100,
                  kde=True,
                  color='skyblue',
                  hist_kws={"linewidth": 15,'alpha':1})
ax.set(xlabel='Uniform ', ylabel='Frequency')

[Text(0, 0.5, 'Frequency'), Text(0.5, 0, 'Uniform ')]

Probability Distributions

Standard Normal Distribuion or Standardization: 

import numpy as np
import pandas as pd
from scipy import stats
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.preprocessing import StandardScaler
%matplotlib inline

dset  = pd.DataFrame([[1],[2],[2],[3],[3],[3],[4],[4],[5]])

print("Mean: ",dset.mean()[0])
print("Standard deviation", dset.std()[0])

Mean:  3.0
Standard deviation 1.224744871391589

density_plot(np.array(dset))

# Applying Standardization.
std_sc = StandardScaler()
zscore_data = std_sc.fit_transform(dset)

/home/tejakummarikuntla/anaconda3/lib/python3.7/site-packages/sklearn/preprocessing/data.py:645: DataConversionWarning: Data with input dtype int64 were all converted to float64 by StandardScaler.
  return self.partial_fit(X, y)
/home/tejakummarikuntla/anaconda3/lib/python3.7/site-packages/sklearn/base.py:464: DataConversionWarning: Data with input dtype int64 were all converted to float64 by StandardScaler.
  return self.fit(X, **fit_params).transform(X)

print("Mean: ",zscore_data.mean())
print("Standard deviation", zscore_data.std())

Mean:  0.0
Standard deviation 1.0

def density_plot(ds):
    f, (ax_box, ax_hist) = plt.subplots(2, sharex=True, gridspec_kw= {"height_ratios": (0.2, 1)})
    mean = np.mean(ds)
    median = np.median(ds)
    mode = stats.mode(ds)[0]
    
    sns.boxplot(ds, ax=ax_box)
    ax_box.axvline(mean, color='r', linestyle='--')
    ax_box.axvline(median, color='g', linestyle='-')
    ax_box.axvline(mode, color='b', linestyle='-')
    
    sns.distplot(ds, hist=True, kde=True, ax=ax_hist,
             color = 'darkblue', 
             hist_kws={'edgecolor':'black'},
             kde_kws={'linewidth': 4})
    ax_hist.axvline(mean, color='r', linestyle='--')
    ax_hist.axvline(median, color='g', linestyle='-')
    ax_hist.axvline(mode, color='b', linestyle='-')
    
    
    plt.legend({'Mean':mean,'Median':median,'Mode':mode})

    ax_box.set(xlabel='')
    plt.show()

density_plot(zscore_data)

type(zscore_data)

numpy.ndarray
np.median(zscore_data)

0.0
stats.mode(dset)[0]

array([[3]])
l=[1,2,4]

dic = {1:"a", 2:"b", 3:"4"}

dic.items()

dict_items([(1, 'a'), (2, 'b'), (3, '4')])
for key,val in dic.items():
    print(key)

1
2
3

lst = np.array([
    [3, 2, 4],
    [6, 7, 8],
    [1, 4, 1]])

two = np.array([
    [1],
    [4],
    [5]
])

lst*two

array([[ 3,  2,  4],
       [24, 28, 32],
       [ 5, 20,  5]])
lst@two

array([[31],
       [74],
       [22]])