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Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
Chi-Squares Test Bivariate Analysis on Categorical Categorical Variables
Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
Supervised Learning vs Unsupervised Learning Reinforcement Learning Generative and Descriminative Models Parametric and Non Parametric model
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Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Understanding Cost Function Understanding Gradient Descent Math Behind Gradient Descent Assumptions of Linear Regression Implement Linear Regression from Scratch Train Linear Regression in Python Implementing Linear Regression in R Diagnosing Residual Plots in Linear Regression Models Generalized Linear Models Introduction to Logistic Regression Odds Ratio Implementing Logistic Regression from Scratch Introduction to Scikit-learn in Python Train Logistic Regression in python Multiclass using Logistic Regression How to use Multinomial and Ordinal Logistic Regression in R ? Challenges with Linear Regression Introduction to Regularisation Implementing Regularisation Ridge Regression Lasso Regression
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naive Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
Understand Basics of Recommendation Engine with Case Study
8 Ways to Improve Accuracy of Machine Learning Models
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
Model Deployment in Android Model Deployment in Iphone

Statistics 101: Beginners Guide to Continuous Probability Distributions

Priyanka
Priyanka Last Updated : 09 Feb, 2021
5 min read

This article was published as a part of the Data Science Blogathon.

Introduction

In the previous post, we have defined Probability Distributions and briefly discussed different Discrete Probability distributions. In this post, we will continue learning about probability distributions through Continuous Probability Distributions.

 

Definition

If you recall from our previous discussion, continuous random variables can take an infinite number of values over a given interval. For example, in the interval [2, 3] there are infinite values between 2 and 3. Continuous distributions are defined by the Probability Density Functions(PDF) instead of Probability Mass Functions. The probability that a continuous random variable is equal to an exact value is always equal to zero. Continuous probabilities are defined over an interval. For instance, P(X = 3) = 0 but P(2.99 < X < 3.01) can be calculated by integrating the PDF over the interval [2.99, 3.01]

 

List of Continuous Probability Distributions

We discuss the most commonly used continuous probability distributions below:

1. Continuous Uniform Distribution

Uniform distribution has both continuous and discrete forms. Here, we discuss the continuous one. This distribution plots the random variables whose values have equal probabilities of occurring. The most common example is flipping a fair die. Here, all 6 outcomes are equally likely to happen. Hence, the probability is constant.

Consider the example where a  = 10 and b = 20, the distribution looks like this:

Continuous Probability Distributions - Uniform

The PDF is given by,

uniform formula

where a is the minimum value and b is the maximum value.

 

2. Normal Distribution

This is the most commonly discussed distribution and most often found in the real world. Many continuous distributions often reach normal distribution given a large enough sample. This has two parameters namely mean and standard deviation.

This distribution has many interesting properties. The mean has the highest probability and all other values are distributed equally on either side of the mean in a symmetric fashion. The standard normal distribution is a special case where the mean is 0 and the standard deviation of 1.

Normal Continuous Probability Distributions

It also follows the empirical formula that 68% of the values are 1 standard deviation away, 95% percent of them are 2 standard deviations away, and 99.7% are 3 standard deviations away from the mean. This property is greatly useful when designing hypothesis tests(https://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/).

The PDF is given by,

normal pdf formula

where μ is the mean of the random variable X and σ is the standard deviation.

 

3. Log-normal Distribution

This distribution is used to plot the random variables whose logarithm values follow a normal distribution. Consider the random variables X and Y. Y = ln(X) is the variable that is represented in this distribution, where ln denotes the natural logarithm of values of X.

log normal distribution

The PDF is given by,

log normal pdf formula

where μ is the mean of Y and σ is the standard deviation of Y.

 

4. Student’s T Distribution

The student’s t distribution is similar to the normal distribution. The difference is that the tails of the distribution are thicker. This is used when the sample size is small and the population variance is not known. This distribution is defined by the degrees of freedom(p) which is calculated as the sample size minus 1(n – 1).

As the sample size increases, degrees of freedom increases the t-distribution approaches the normal distribution and the tails become narrower and the curve gets closer to the mean. This distribution is used to test estimates of the population mean when the sample size is less than 30 and population variance is unknown. The sample variance/standard deviation is used to calculate the t-value.

Continuous Probability Distributions - t-distribution

The PDF is given by,

t-distribution formula

where p is the degrees of freedom and Γ is the gamma function. Check this link for a brief description of the gamma function.

The t-statistic used in hypothesis testing is calculated as follows,

Continuous Probability Distributions t formula

where x̄ is the sample mean, μ the population mean and s is the sample variance.

 

5. Chi-square Distribution

This distribution is equal to the sum of squares of p normal random variables. p is the number of degrees of freedom. Like the t-distribution, as the degrees of freedom increase, the distribution gradually approaches the normal distribution. Below is a chi-square distribution with three degrees of freedom.

Continuous Probability Distributions chi

The PDF is given by,

chi square pdf

where p is the degrees of freedom and Γ is the gamma function.

The chi-square value is calculated as follows:

chi formula

where o is the observed value and E represents the expected value. This is used in hypothesis testing to draw inferences about the population variance of normal distributions.

 

6. Exponential Distribution

Recall the discrete probability distribution we have discussed in the Discrete Probability post. In the Poisson distribution, we took the example of calls received by the customer care center. In that example, we considered the average number of calls per hour. Now, in this distribution, the time between successive calls is explained.

Exponential Distribution formula

The exponential distribution can be seen as an inverse of the Poisson distribution. The events in consideration are independent of each other.

The PDF is given by,

exponential pdf

where λ is the rate parameter. λ = 1/(average time between events).

To conclude, we have very briefly discussed different continuous probability distributions in this post. Feel free to add any comments or suggestions below.

 

About Me

I am Priyanka Madiraju, a former software engineer, working on transitioning into Data Science. I am a master’s student in Data Science. Please feel free to connect with me on https://www.linkedin.com/in/priyanka-madiraju

The media shown in this article are not owned by Analytics Vidhya and is used at the Author’s discretion.

Priyanka
Priyanka

I am a former software engineer with 6 years of work experience. I am pursuing my Masters in Data Science student @ TU Dortmund. I write about my areas of interest regularly on LinkedIn and Medium. Follow me for more technical content.

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Responses From Readers

Clear

Andrew James
Andrew James

Hi, I also read your previous post and very much impressed with the knowledge. I really thank you to share both articles. Regards

123

GIDEON OGIDI-GH
GIDEON OGIDI-GH

God richly bless you Brother for such a brief explanation. but please if you could add real life examples to each distribution type for more better understanding. Thank you Snr

123

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