Continuous Distributions
Normal distribution with python code
Utilities Code Functions
This code was run in a Jupyter Notebook
Python Here Import Libraries
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import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
Python Here norm_sim()
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def norm_sim(frame,
mu=0,
sd=1,
count_sampes=10000,
):
"""
Parameters:
------------
mu: center location of the normal distribution
sd: standard deviation
"""
#_______________________________________________
# GENERATING RANDOM VALUES FROM NORMAL DISTRIBUTION
rng = np.random.default_rng(seed=0) # adding a seed.
random_numbers = rng.normal(loc=mu,
scale=sd,
size=int(((count_sampes)/50)*(frame+1)),
)
#__________________________________________________
# CREATING A NORMAL DISTRIBUTION PDF WILL HELP TO
# VIZUALLY VERIFY IF THESE VALUES COME FROM A NOORMAL DISTRIBUTION
x = np.linspace(min(random_numbers), max(random_numbers))
pdf = norm.pdf(x=x, # list-like set of x value to calculate the PDF
loc=mu, # location means where is the center, it's the mean.
scale=sd, # here scale refers to standard deviation
)
#_____________________________________________
# VIZ
# Clear current axes
plt.cla() # Clear current axes
plt.clf() # Clear the current figure
# plt.figure(figsize=(12,8)) # this should be outsite of FuncAnimation()
ax = plt.subplot(121) # Create subplot
plt.scatter(np.arange(len(random_numbers)), random_numbers, c='#1f77b4', alpha=0.1,)
plt.xlabel("Sample count")
plt.title(rf"Generation of random values from $n(x, \mu={mu},\sigma={sd})$")
plt.gca().set_facecolor('lightgrey')
plt.ylim(-4,4)
plt.xlim(0, count_sampes)
ax = plt.subplot(122) # Create subplot
plt.hist(random_numbers,
orientation='horizontal',
bins=100,
cumulative=False,
density=True,
color='#1f77b4',
)
plt.plot(pdf, x, 'r-', lw=2, alpha=.5, label="PDF")
plt.title("Histogram of the generated random values")
plt.ylim(-4,4)
plt.xlim(0, .5)
plt.xlabel("density")
plt.legend(framealpha=0)
plt.gcf().set_facecolor('lightgrey')
plt.gca().set_facecolor('lightgrey')
plt.suptitle("Normal Distribution simulation")
plt.tight_layout()
Normal Distribution
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution where values are symmetrically distributed mostly around the mean, forming a bell shape. It is characterized by two parameters: the mean and the standard deviation, which determine the distribution’s appearance. A continuous random variable that exhibits this bell-shaped behavior is called a normal random variable.
Probability Density Function:
\[f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^2 }\]Python VIZ Here
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#_______________________________________________
# PARAMETERS
# Define the range
x = np.linspace(-5, 5, 1000) # array of values for "x" in f(x)
mu = 0 # mean
sd = 1 # standard deviation (sigma)
#_______________________________________________
# PROBABILITY DENSITY FUNCTION TO GENERATE BELL SHAPE
pdf = norm.pdf(x=x, # list-like set of x value to calculate the PDF
loc=mu, # location means where is the center, it's the mean.
scale=sd, # here scale refers to standard deviation
)
#_______________________________________________
# VIZ
plt.figure(figsize=(8, 4),
facecolor='lightgrey'
)
plt.plot(x, pdf, label='PDF', color='#1f77b4')
plt.title('Probability Density Function (PMF) of Normal Distribution')
plt.xlabel('x')
plt.ylabel('Probability Density')
plt.legend(framealpha=0 # transparent backround label.
)
plt.gca().set_facecolor('lightgrey')
plt.grid(color='gray', linestyle="--", zorder=0 )
# plt.savefig('PDF_nrormal_distribution.jpg',
# dpi=300, # useful for papers publications.
# bbox_inches='tight')
plt.show()
Analytical Thought Process
- Consider the bell-shaped curve as a potential model to describe observed data distribution.
- Apply statistical techniques to test if the observed data fit the bell-shaped curve.
- If there is a good fit, we may conclude we have a normal random variable, which allows for parametric analysis.
Notes:
scipy.stats.norm.pdf()
return Probability Density Function (PDF) for normal distributions.
Python VIZ Here
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%matplotlib notebook
fig = plt.figure(figsize=(12,8))
ani = FuncAnimation(fig,
func=norm_sim,
frames=range(50),
interval=200, # ms
)
plt.gcf().set_facecolor('lightgrey')
# ani.save('normal_distribution_simulation.gif', writer='pillow', fps=5)
HTML(ani.to_jshtml())
Notes:
np.random.normal()
generate random values from a normal distributions.
Concepts
Probability Density Function
- The probability of a continuous random variable X assuming any exact value is 0. However, the value of the PDF at a particular point can indeed be greater than 0. This means that the PDF value does not equate to a probability but rather indicates the relative likelihood of X being near that value. To obtain probability from the PDF, we need to calculate the area under the curve between two points of X.
References
- Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2007). Probability and statistics for engineers and scientists. New York: Macmillan.
This post is licensed under CC BY 4.0 by the author.