Probability distributions play a crucial role in mathematics, particularly in the fields of probability theory and statistics. They represent the values of a variable and the probabilities associated with different outcomes in an experiment. In the realm of data science and machine learning, probability distributions hold significant importance, especially when it comes to understanding the properties of data. In this article, we will delve into the popular probability distributions, explore their differences, and learn how to visualize them using Python.
Exploring Probability Distributions
What is a Probability Distribution?
In probability theory and statistics, a probability distribution is a representation of the values of a variable along with their associated probabilities. In machine learning and data science, probability distributions are extensively utilized. The study of probability distributions becomes essential in machine learning as models often need to learn the uncertainty within the data. By categorizing probability distributions and data, we can further explore the subject of probability distribution in machine learning.
Categorizing Data Types
In machine learning, we often encounter different formats of data. Datasets can be considered as differentiated samples from a population, and finding patterns within these samples is crucial for making predictions about the entire dataset or population. Data elements can be classified into two types:
- Numerical:
- Discrete: This type of numerical data can only take specific values, such as the number of apples in a basket or the number of people in a team.
- Continuous: This type of numerical data can take real or fractional values, like the height or width of a tree.
- Categorical:
- This type of data includes categories such as gender or state.
Analyzing the dataset using discrete random variables allows us to calculate the probability mass function, while continuous random variables involve the calculation of the probability density function.
Elements of the Probability Distribution
There are two fundamental functions used to obtain probability distributions:
- Probability Mass Function (PMF): The PMF gives the probability that a discrete random variable is equal to a specific value. It represents a discrete probability distribution.
- Probability Density Function (PDF): The PDF represents the density of a continuous random variable within a specific range of values. It is associated with continuous probability distributions.
Discrete Probability Distributions
Under the umbrella of discrete probability distributions, there are several popular distributions that can be used in Python. Let’s take a look at them:
1. Binomial Distribution
The binomial distribution summarizes the likelihood that a variable will take one of two values based on a set of pre-assumed parameters. This distribution is commonly used in scenarios involving sequences of experiments where the outcomes are binary, such as yes/no or positive/negative. These experiments are known as Bernoulli trials or Bernoulli experiments. The probability mass function for the binomial distribution is as follows:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Here, k belongs to the set {0, 1, …, n} and 0 <= p <= 1. To generate binomial discrete random variables using Python, you can utilize the following code:
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
for prob in range(3, 10, 3):
x = np.arange(0, 25)
binom = stats.binom.pmf(x, 20, 0.1 * prob)
plt.plot(x, binom, '-o', label="p = {:f}".format(0.1 * prob))
plt.xlabel('Random Variable', fontsize=12)
plt.ylabel('Probability', fontsize=12)
plt.title("Binomial Distribution varying p")
plt.legend()
plt.show()
2. Poisson Distribution
The Poisson distribution is a subcategory of discrete probability distribution that represents the probability of a certain number of events occurring within a fixed range of time. It is commonly used when the variable of interest in the data is discrete. The probability mass function for the Poisson distribution is given by:
P(X = k) = (e^(-λ) * λ^k) / k!
Here, k >= 0. To visualize the Poisson distribution in Python, you can use the following code:
for lambd in range(2, 8, 2):
n = np.arange(0, 10)
poisson = stats.poisson.pmf(n, lambd)
plt.plot(n, poisson, '-o', label="λ = {:f}".format(lambd))
plt.xlabel('Number of Events', fontsize=12)
plt.ylabel('Probability', fontsize=12)
plt.title("Poisson Distribution varying λ")
plt.legend()
plt.show()
Continuous Probability Distributions
Moving on to continuous probability distributions, let’s explore some popular distributions and how they can be utilized in Python.
1. Normal Distribution
The normal distribution, also known as the Gaussian distribution, represents the probability distribution for a real-valued random variable. It is widely used in various statistical analyses. The probability density function for the normal distribution is given by:
f(x) = (1 / sqrt(2πσ^2)) * e^(-((x-μ)^2 / (2σ^2)))
Here, μ represents the mean and σ represents the standard deviation. To represent the normal distribution in Python, you can utilize the following code:
from seaborn.palettes import color_palette
n = np.arange(-70, 70)
norm = stats.norm.pdf(n, 0, 10)
plt.plot(n, norm)
plt.xlabel('Distribution', fontsize=12)
plt.ylabel('Probability', fontsize=12)
plt.title("Normal Distribution of x")
plt.show()
2. Uniform Distribution
The uniform distribution is a subcategory of continuous probability distribution where all events have similar probabilities of occurring. It is often associated with scenarios such as rolling a fair dice, where each face has an equal probability of occurring. The probability density function for the uniform distribution is given by:
f(x) = 1 / (b - a)
Here, a and b represent the range of values. To represent the distribution of probabilities when rolling a fair dice, you can use the following code:
probs = np.full((6), 1/6)
face = [1, 2, 3, 4, 5, 6]
plt.bar(face, probs)
plt.ylabel('Probability', fontsize=12)
plt.xlabel('Dice Roll Outcome', fontsize=12)
plt.title('Fair Dice Uniform Distribution', fontsize=12)
axes = plt.gca()
axes.set_ylim([0, 1])
plt.show()
Final Words
In this comprehensive article, we have explored the concept of probability distributions. We have discussed how to categorize them based on data types and their elements. Additionally, we have delved into discrete and continuous probability distributions, exploring their sub-categories and providing visualizations using Python.
Understanding probability distributions is essential in various fields, especially in data science and machine learning. By grasping the concepts and visualizing the distributions, you can gain valuable insights into the behavior of data and make informed decisions based on probability.