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Eigenvalues and Eigenvectors
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System of Linear Equations in three variables using Cramer's Rule

Last Updated : 12 Aug, 2024
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Cramer's rule: In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknown variables. It expresses the solution in terms of the determinants of the coefficient matrix and of matrices obtained from it by replacing one column by the column vector of the right-hand-sides of the equations. Cramer's rule is computationally inefficient for systems of more than two or three equations.
Suppose we have to solve these equations: 

a1x + b1y + c1z = d1 
a2x + b2y + c2z = d2 
a3x + b3y + c3z = d3 

Following the Cramer's Rule, first find the determinant values of all four matrices. 

D = \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3\\ \end{vmatrix}       

D_1 = \begin{vmatrix} d_1 & b_1 & c_1\\ d_2 & b_2 & c_2\\ d_3 & b_3 & c_3\\ \end{vmatrix}       

D_2 = \begin{vmatrix} a_1 & d_1 & c_1\\ a_2 & d_2 & c_2\\ a_3 & d_3 & c_3\\ \end{vmatrix}       

D_3 = \begin{vmatrix} a_1 & b_1 & d_1\\ a_2 & b_2 & d_2\\ a_3 & b_3 & d_3\\ \end{vmatrix}    

There are 2 cases: 

  • Case I : When D ≠ 0 In this case we have,  
    • x = D1/D 
    • y = D2/D 
    • z = D3/D                
    • Hence unique value of x, y, z will be obtained. 
  • Case II : When D = 0         
    • When at least one of D1, D2 and D3 is non zero: Then no solution is possible and hence system of equations will be inconsistent. 
    • When D = 0 and D1 = D2 = D3 = 0: Then the system of equations will be consistent and it will have infinitely many solutions.  

Example: 

Consider the following system of linear equations. 
[2x - y + 3z = 9], [x + y + z = 6], [x - y + z = 2] 

D = \begin{vmatrix} 2 & -1 & 3\\ 1 & 1 & 1\\ 1 & -1 & 1\\ \end{vmatrix}         D_1 = \begin{vmatrix} 9 & -1 & 3\\ 6 & 1 & 1\\ 2 & -1 & 1\\ \end{vmatrix}         D_2 = \begin{vmatrix} 2 & 9 & 3\\ 1 & 6 & 1\\ 1 & 2 & 1\\ \end{vmatrix}         D_3 = \begin{vmatrix} 2 & -1 & 9\\ 1 & 1 & 6\\ 1 & -1 & 2\\ \end{vmatrix}      

[x = D1/D = 1], [y = D2/D = 2], [z = D3/D = 3]  

Below is the implementation. 

C++ 
// CPP program to calculate solutions of linear // equations using cramer's rule #include <bits/stdc++.h> using namespace std;  // This functions finds the determinant of Matrix double determinantOfMatrix(double mat[3][3]) {     double ans;     ans = mat[0][0] * (mat[1][1] * mat[2][2] - mat[2][1] * mat[1][2])           - mat[0][1] * (mat[1][0] * mat[2][2] - mat[1][2] * mat[2][0])           + mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);     return ans; }  // This function finds the solution of system of // linear equations using cramer's rule void findSolution(double coeff[3][4]) {     // Matrix d using coeff as given in cramer's rule     double d[3][3] = {         { coeff[0][0], coeff[0][1], coeff[0][2] },         { coeff[1][0], coeff[1][1], coeff[1][2] },         { coeff[2][0], coeff[2][1], coeff[2][2] },     };     // Matrix d1 using coeff as given in cramer's rule     double d1[3][3] = {         { coeff[0][3], coeff[0][1], coeff[0][2] },         { coeff[1][3], coeff[1][1], coeff[1][2] },         { coeff[2][3], coeff[2][1], coeff[2][2] },     };     // Matrix d2 using coeff as given in cramer's rule     double d2[3][3] = {         { coeff[0][0], coeff[0][3], coeff[0][2] },         { coeff[1][0], coeff[1][3], coeff[1][2] },         { coeff[2][0], coeff[2][3], coeff[2][2] },     };     // Matrix d3 using coeff as given in cramer's rule     double d3[3][3] = {         { coeff[0][0], coeff[0][1], coeff[0][3] },         { coeff[1][0], coeff[1][1], coeff[1][3] },         { coeff[2][0], coeff[2][1], coeff[2][3] },     };      // Calculating Determinant of Matrices d, d1, d2, d3     double D = determinantOfMatrix(d);     double D1 = determinantOfMatrix(d1);     double D2 = determinantOfMatrix(d2);     double D3 = determinantOfMatrix(d3);     printf("D is : %lf \n", D);     printf("D1 is : %lf \n", D1);     printf("D2 is : %lf \n", D2);     printf("D3 is : %lf \n", D3);      // Case 1     if (D != 0) {         // Coeff have a unique solution. Apply Cramer's Rule         double x = D1 / D;         double y = D2 / D;         double z = D3 / D; // calculating z using cramer's rule         printf("Value of x is : %lf\n", x);         printf("Value of y is : %lf\n", y);         printf("Value of z is : %lf\n", z);     }     // Case 2     else {         if (D1 == 0 && D2 == 0 && D3 == 0)             printf("Infinite solutions\n");         else if (D1 != 0 || D2 != 0 || D3 != 0)             printf("No solutions\n");     } }  // Driver Code int main() {      // storing coefficients of linear equations in coeff matrix     double coeff[3][4] = {         { 2, -1, 3, 9 },         { 1, 1, 1, 6 },         { 1, -1, 1, 2 },     };      findSolution(coeff);     return 0; }
Java 
// Java program to calculate solutions of linear // equations using cramer's rule class GFG {  // This functions finds the determinant of Matrix static double determinantOfMatrix(double mat[][]) {     double ans;     ans = mat[0][0] * (mat[1][1] * mat[2][2] - mat[2][1] * mat[1][2])         - mat[0][1] * (mat[1][0] * mat[2][2] - mat[1][2] * mat[2][0])         + mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);     return ans; }  // This function finds the solution of system of // linear equations using cramer's rule static void findSolution(double coeff[][]) {     // Matrix d using coeff as given in cramer's rule     double d[][] = {         { coeff[0][0], coeff[0][1], coeff[0][2] },         { coeff[1][0], coeff[1][1], coeff[1][2] },         { coeff[2][0], coeff[2][1], coeff[2][2] },     };          // Matrix d1 using coeff as given in cramer's rule     double d1[][] = {         { coeff[0][3], coeff[0][1], coeff[0][2] },         { coeff[1][3], coeff[1][1], coeff[1][2] },         { coeff[2][3], coeff[2][1], coeff[2][2] },     };          // Matrix d2 using coeff as given in cramer's rule     double d2[][] = {         { coeff[0][0], coeff[0][3], coeff[0][2] },         { coeff[1][0], coeff[1][3], coeff[1][2] },         { coeff[2][0], coeff[2][3], coeff[2][2] },     };          // Matrix d3 using coeff as given in cramer's rule     double d3[][] = {         { coeff[0][0], coeff[0][1], coeff[0][3] },         { coeff[1][0], coeff[1][1], coeff[1][3] },         { coeff[2][0], coeff[2][1], coeff[2][3] },     };      // Calculating Determinant of Matrices d, d1, d2, d3     double D = determinantOfMatrix(d);     double D1 = determinantOfMatrix(d1);     double D2 = determinantOfMatrix(d2);     double D3 = determinantOfMatrix(d3);     System.out.printf("D is : %.6f \n", D);     System.out.printf("D1 is : %.6f \n", D1);     System.out.printf("D2 is : %.6f \n", D2);     System.out.printf("D3 is : %.6f \n", D3);      // Case 1     if (D != 0)      {         // Coeff have a unique solution. Apply Cramer's Rule         double x = D1 / D;         double y = D2 / D;         double z = D3 / D; // calculating z using cramer's rule         System.out.printf("Value of x is : %.6f\n", x);         System.out.printf("Value of y is : %.6f\n", y);         System.out.printf("Value of z is : %.6f\n", z);     }          // Case 2     else      {         if (D1 == 0 && D2 == 0 && D3 == 0)             System.out.printf("Infinite solutions\n");         else if (D1 != 0 || D2 != 0 || D3 != 0)             System.out.printf("No solutions\n");     } }  // Driver Code public static void main(String[] args) {     // storing coefficients of linear     // equations in coeff matrix     double coeff[][] = {{ 2, -1, 3, 9 },                         { 1, 1, 1, 6 },                         { 1, -1, 1, 2 }};     findSolution(coeff);     } }   // This code is contributed by PrinciRaj1992
Python 
# Python3 program to calculate  # solutions of linear equations  # using cramer's rule  # This functions finds the  # determinant of Matrix def determinantOfMatrix(mat):      ans = (mat[0][0] * (mat[1][1] * mat[2][2] -                          mat[2][1] * mat[1][2]) -             mat[0][1] * (mat[1][0] * mat[2][2] -                         mat[1][2] * mat[2][0]) +             mat[0][2] * (mat[1][0] * mat[2][1] -                          mat[1][1] * mat[2][0]))     return ans  # This function finds the solution of system of # linear equations using cramer's rule def findSolution(coeff):      # Matrix d using coeff as given in      # cramer's rule     d = [[coeff[0][0], coeff[0][1], coeff[0][2]],          [coeff[1][0], coeff[1][1], coeff[1][2]],          [coeff[2][0], coeff[2][1], coeff[2][2]]]          # Matrix d1 using coeff as given in      # cramer's rule     d1 = [[coeff[0][3], coeff[0][1], coeff[0][2]],           [coeff[1][3], coeff[1][1], coeff[1][2]],           [coeff[2][3], coeff[2][1], coeff[2][2]]]          # Matrix d2 using coeff as given in      # cramer's rule     d2 = [[coeff[0][0], coeff[0][3], coeff[0][2]],           [coeff[1][0], coeff[1][3], coeff[1][2]],           [coeff[2][0], coeff[2][3], coeff[2][2]]]          # Matrix d3 using coeff as given in      # cramer's rule     d3 = [[coeff[0][0], coeff[0][1], coeff[0][3]],           [coeff[1][0], coeff[1][1], coeff[1][3]],           [coeff[2][0], coeff[2][1], coeff[2][3]]]      # Calculating Determinant of Matrices      # d, d1, d2, d3     D = determinantOfMatrix(d)     D1 = determinantOfMatrix(d1)     D2 = determinantOfMatrix(d2)     D3 = determinantOfMatrix(d3)          print("D is : ", D)     print("D1 is : ", D1)     print("D2 is : ", D2)     print("D3 is : ", D3)      # Case 1     if (D != 0):                # Coeff have a unique solution.          # Apply Cramer's Rule         x = D1 / D         y = D2 / D                  # calculating z using cramer's rule         z = D3 / D                    print("Value of x is : ", x)         print("Value of y is : ", y)         print("Value of z is : ", z)      # Case 2     else:         if (D1 == 0 and D2 == 0 and              D3 == 0):             print("Infinite solutions")         elif (D1 != 0 or D2 != 0 or                D3 != 0):             print("No solutions")  # Driver Code if __name__ == "__main__":      # storing coefficients of linear      # equations in coeff matrix     coeff = [[2, -1, 3, 9],              [1, 1, 1, 6],              [1, -1, 1, 2]]      findSolution(coeff)  # This code is contributed by Chitranayal
C# 
// C# program to calculate solutions of linear // equations using cramer's rule using System;  class GFG {  // This functions finds the determinant of Matrix static double determinantOfMatrix(double [,]mat) {     double ans;     ans = mat[0,0] * (mat[1,1] * mat[2,2] - mat[2,1] * mat[1,2])         - mat[0,1] * (mat[1,0] * mat[2,2] - mat[1,2] * mat[2,0])         + mat[0,2] * (mat[1,0] * mat[2,1] - mat[1,1] * mat[2,0]);     return ans; }  // This function finds the solution of system of // linear equations using cramer's rule static void findSolution(double [,]coeff) {     // Matrix d using coeff as given in cramer's rule     double [,]d = {         { coeff[0,0], coeff[0,1], coeff[0,2] },         { coeff[1,0], coeff[1,1], coeff[1,2] },         { coeff[2,0], coeff[2,1], coeff[2,2] },     };          // Matrix d1 using coeff as given in cramer's rule     double [,]d1 = {         { coeff[0,3], coeff[0,1], coeff[0,2] },         { coeff[1,3], coeff[1,1], coeff[1,2] },         { coeff[2,3], coeff[2,1], coeff[2,2] },     };          // Matrix d2 using coeff as given in cramer's rule     double [,]d2 = {         { coeff[0,0], coeff[0,3], coeff[0,2] },         { coeff[1,0], coeff[1,3], coeff[1,2] },         { coeff[2,0], coeff[2,3], coeff[2,2] },     };          // Matrix d3 using coeff as given in cramer's rule     double [,]d3 = {         { coeff[0,0], coeff[0,1], coeff[0,3] },         { coeff[1,0], coeff[1,1], coeff[1,3] },         { coeff[2,0], coeff[2,1], coeff[2,3] },     };      // Calculating Determinant of Matrices d, d1, d2, d3     double D = determinantOfMatrix(d);     double D1 = determinantOfMatrix(d1);     double D2 = determinantOfMatrix(d2);     double D3 = determinantOfMatrix(d3);     Console.Write("D is : {0:F6} \n", D);     Console.Write("D1 is : {0:F6} \n", D1);     Console.Write("D2 is : {0:F6} \n", D2);     Console.Write("D3 is : {0:F6} \n", D3);      // Case 1     if (D != 0)      {         // Coeff have a unique solution. Apply Cramer's Rule         double x = D1 / D;         double y = D2 / D;         double z = D3 / D; // calculating z using cramer's rule         Console.Write("Value of x is : {0:F6}\n", x);         Console.Write("Value of y is : {0:F6}\n", y);         Console.Write("Value of z is : {0:F6}\n", z);     }          // Case 2     else     {         if (D1 == 0 && D2 == 0 && D3 == 0)             Console.Write("Infinite solutions\n");         else if (D1 != 0 || D2 != 0 || D3 != 0)             Console.Write("No solutions\n");     } }  // Driver Code public static void Main() {     // storing coefficients of linear     // equations in coeff matrix     double [,]coeff = {{ 2, -1, 3, 9 },                         { 1, 1, 1, 6 },                         { 1, -1, 1, 2 }};     findSolution(coeff);     } }   // This code is contributed by 29AjayKumar
JavaScript 
<script> // Javascript program to calculate solutions of linear // equations using cramer's rule  // This functions finds the determinant of Matrix function determinantOfMatrix(mat) {     let ans;     ans = mat[0][0] * (mat[1][1] * mat[2][2] - mat[2][1] * mat[1][2])         - mat[0][1] * (mat[1][0] * mat[2][2] - mat[1][2] * mat[2][0])         + mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);     return ans;      }  // This function finds the solution of system of // linear equations using cramer's rule function findSolution(coeff) { // Matrix d using coeff as given in cramer's rule     let d = [[coeff[0][0], coeff[0][1], coeff[0][2]],          [coeff[1][0], coeff[1][1], coeff[1][2]],          [coeff[2][0], coeff[2][1], coeff[2][2]]];          // Matrix d1 using coeff as given in cramer's rule     let d1 = [[coeff[0][3], coeff[0][1], coeff[0][2]],           [coeff[1][3], coeff[1][1], coeff[1][2]],           [coeff[2][3], coeff[2][1], coeff[2][2]]];                // Matrix d2 using coeff as given in cramer's rule           let d2 = [[coeff[0][0], coeff[0][3], coeff[0][2]],           [coeff[1][0], coeff[1][3], coeff[1][2]],           [coeff[2][0], coeff[2][3], coeff[2][2]]];          // Matrix d3 using coeff as given in cramer's rule     let d3 = [[coeff[0][0], coeff[0][1], coeff[0][3]],           [coeff[1][0], coeff[1][1], coeff[1][3]],           [coeff[2][0], coeff[2][1], coeff[2][3]]];          // Calculating Determinant of Matrices d, d1, d2, d3     let D = determinantOfMatrix(d);     let D1 = determinantOfMatrix(d1);     let D2 = determinantOfMatrix(d2);     let D3 = determinantOfMatrix(d3);          document.write("D is :  ", D.toFixed(6)+"<br>");     document.write("D1 is : ", D1.toFixed(6)+"<br>");     document.write("D2 is : ", D2.toFixed(6)+"<br>");     document.write("D3 is : ", D3.toFixed(6)+"<br>");          // Case 1     if (D != 0)     {         // Coeff have a unique solution. Apply Cramer's Rule         let x = D1 / D;         let y = D2 / D;         let z = D3 / D; // calculating z using cramer's rule         document.write("Value of x is : ", x.toFixed(6)+"<br>");         document.write("Value of y is : ", y.toFixed(6)+"<br>");         document.write("Value of z is : ", z.toFixed(6)+"<br>");     }           // Case 2     else     {         if (D1 == 0 && D2 == 0 && D3 == 0)             document.write("Infinite solutions\n");         else if (D1 != 0 || D2 != 0 || D3 != 0)             document.write("No solutions\n");     } }  // Driver Code let coeff = [[2, -1, 3, 9],              [1, 1, 1, 6],              [1, -1, 1, 2]]   findSolution(coeff);               // This code is contributed by avanitrachhadiya2155 </script>

Output
D is : -2.000000  D1 is : -2.000000  D2 is : -4.000000  D3 is : -6.000000  Value of x is : 1.000000 Value of y is : 2.000000 Value of z is : 3.000000

Time complexity: O(1) 
Auxiliary space: O(1)


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    Statistics for Machine Learning

    Descriptive Statistic
    Statistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
    5 min read
    Measures of Central Tendency
    Usually, frequency distribution and graphical representation are used to depict a set of raw data to attain meaningful conclusions from them. However, sometimes, these methods fail to convey a proper and clear picture of the data as expected. Therefore, some measures, also known as Measures of Centr
    5 min read
    Measures of Dispersion | Types, Formula and Examples
    Measures of Dispersion are used to represent the scattering of data. These are the numbers that show the various aspects of the data spread across multiple parameters.Let's learn about the measure of dispersion in statistics, its types, formulas, and examples in detail.Dispersion in StatisticsDisper
    9 min read
    Mean, Variance and Standard Deviation
    Mean, Variance and Standard Deviation are fundamental concepts in statistics and engineering mathematics, essential for analyzing and interpreting data. These measures provide insights into data's central tendency, dispersion, and spread, which are crucial for making informed decisions in various en
    10 min read
    Calculate the average, variance and standard deviation in Python using NumPy
    Numpy in Python is a general-purpose array-processing package. It provides a high-performance multidimensional array object and tools for working with these arrays. It is the fundamental package for scientific computing with Python. Numpy provides very easy methods to calculate the average, variance
    5 min read
    Random Variable
    Random variable is a fundamental concept in statistics that bridges the gap between theoretical probability and real-world data. A Random variable in statistics is a function that assigns a real value to an outcome in the sample space of a random experiment. For example: if you roll a die, you can a
    10 min read
    Difference between Parametric and Non-Parametric Methods
    Statistical analysis plays a crucial role in understanding and interpreting data across various disciplines. Two prominent approaches in statistical analysis are Parametric and Non-Parametric Methods. While both aim to draw inferences from data, they differ in their assumptions and underlying princi
    8 min read
    Probability Distribution - Function, Formula, Table
    A probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a frequency distribution shows
    15+ min read
    Confidence Interval
    A Confidence Interval (CI) is a range of values that contains the true value of something we are trying to measure like the average height of students or average income of a population.Instead of saying: “The average height is 165 cm.”We can say: “We are 95% confident the average height is between 1
    7 min read
    Covariance and Correlation
    Covariance and correlation are the two key concepts in Statistics that help us analyze the relationship between two variables. Covariance measures how two variables change together, indicating whether they move in the same or opposite directions. Relationship between Independent and dependent variab
    5 min read
    Program to Find Correlation Coefficient
    The correlation coefficient is a statistical measure that helps determine the strength and direction of the relationship between two variables. It quantifies how changes in one variable correspond to changes in another. This coefficient, sometimes referred to as the cross-correlation coefficient, al
    8 min read
    Robust Correlation
    Correlation is a statistical tool that is used to analyze and measure the degree of relationship or degree of association between two or more variables. There are generally three types of correlation: Positive correlation: When we increase the value of one variable, the value of another variable inc
    8 min read
    Normal Probability Plot
    The probability plot is a way of visually comparing the data coming from different distributions. These data can be of empirical dataset or theoretical dataset. The probability plot can be of two types:P-P plot: The (Probability-to-Probability) p-p plot is the way to visualize the comparing of cumul
    3 min read
    Quantile Quantile plots
    The quantile-quantile( q-q plot) plot is a graphical method for determining if a dataset follows a certain probability distribution or whether two samples of data came from the same population or not. Q-Q plots are particularly useful for assessing whether a dataset is normally distributed or if it
    8 min read
    True Error vs Sample Error
    True Error The true error can be said as the probability that the hypothesis will misclassify a single randomly drawn sample from the population. Here the population represents all the data in the world. Let's consider a hypothesis h(x) and the true/target function is f(x) of population P. The proba
    3 min read
    Bias-Variance Trade Off - Machine Learning
    It is important to understand prediction errors (bias and variance) when it comes to accuracy in any machine-learning algorithm. There is a tradeoff between a model’s ability to minimize bias and variance which is referred to as the best solution for selecting a value of Regularization constant. A p
    3 min read
    Hypothesis Testing
    Hypothesis testing compares two opposite ideas about a group of people or things and uses data from a small part of that group (a sample) to decide which idea is more likely true. We collect and study the sample data to check if the claim is correct.Hypothesis TestingFor example, if a company says i
    9 min read
    T-test
    After learning about the Z-test we now move on to another important statistical test called the t-test. While the Z-test is useful when we know the population variance. The t-test is used to compare the averages of two groups to see if they are significantly different from each other. Suppose you wa
    6 min read
    Paired T-Test - A Detailed Overview
    Student’s t-test or t-test is the statistical method used to determine if there is a difference between the means of two samples. The test is often performed to find out if there is any sampling error or unlikeliness in the experiment. This t-test is further divided into 3 types based on your data a
    5 min read
    P-value in Machine Learning
    P-value helps us determine how likely it is to get a particular result when the null hypothesis is assumed to be true. It is the probability of getting a sample like ours or more extreme than ours if the null hypothesis is correct. Therefore, if the null hypothesis is assumed to be true, the p-value
    6 min read
    F-Test in Statistics
    F test is a statistical test that is used in hypothesis testing that determines whether the variances of two samples are equal or not. The article will provide detailed information on f test, f statistic, its calculation, critical value and how to use it to test hypotheses. To understand F test firs
    6 min read
    Z-test : Formula, Types, Examples
    A Z-test is a type of hypothesis test that compares the sample’s average to the population’s average and calculates the Z-score and tells us how much the sample average is different from the population average by looking at how much the data normally varies. It is particularly useful when the sample
    8 min read
    Residual Leverage Plot (Regression Diagnostic)
    In linear or multiple regression, it is not enough to just fit the model into the dataset. But, it may not give the desired result. To apply the linear or multiple regression efficiently to the dataset. There are some assumptions that we need to check on the dataset that made linear/multiple regress
    5 min read
    Difference between Null and Alternate Hypothesis
    Hypothesis is a statement or an assumption that may be true or false. There are six types of hypotheses mainly the Simple hypothesis, Complex hypothesis, Directional hypothesis, Associative hypothesis, and Null hypothesis. Usually, the hypothesis is the start point of any scientific investigation, I
    3 min read
    Mann and Whitney U test
    Mann and Whitney's U-test or Wilcoxon rank-sum testis the non-parametric statistic hypothesis test that is used to analyze the difference between two independent samples of ordinal data. In this test, we have provided two randomly drawn samples and we have to verify whether these two samples is from
    5 min read
    Wilcoxon Signed Rank Test
    The Wilcoxon Signed Rank Test is a non-parametric statistical test used to compare two related groups. It is often applied when the assumptions for the paired t-test (such as normality) are not met. This test evaluates whether there is a significant difference between two paired observations, making
    5 min read
    Kruskal Wallis Test
    The Kruskal-Wallis test (H test) is a nonparametric statistical test used to compare three or more independent groups to determine if there are statistically significant differences between them. It is an extension of the Mann-Whitney U test, which is used for comparing two groups.Unlike the one-way
    4 min read
    Friedman Test
    The Friedman Test is a non-parametric statistical test used to detect differences in treatments across multiple test attempts. It is often used when the data is in the form of rankings or ordinal data, and when you have more than two related groups or repeated measures. The Friedman test is the non-
    6 min read
    Probability Class 10 Important Questions
    Probability is a fundamental concept in mathematics for measuring of chances of an event happening By assigning numerical values to the chances of different outcomes, probability allows us to model, analyze, and predict complex systems and processes.Probability Formulas for Class 10 It says the poss
    4 min read

    Probability and Probability Distributions

    Mathematics - Law of Total Probability
    Probability theory is the branch of mathematics concerned with the analysis of random events. It provides a framework for quantifying uncertainty, predicting outcomes, and understanding random phenomena. In probability theory, an event is any outcome or set of outcomes from a random experiment, and
    12 min read
    Bayes's Theorem for Conditional Probability
    Bayes's Theorem for Conditional Probability: Bayes's Theorem is a fundamental result in probability theory that describes how to update the probabilities of hypotheses when given evidence. Named after the Reverend Thomas Bayes, this theorem is crucial in various fields, including engineering, statis
    9 min read
    Uniform Distribution in Data Science
    Uniform Distribution also known as the Rectangular Distribution is a type of Continuous Probability Distribution where all outcomes in a given interval are equally likely. Unlike Normal Distribution which have varying probabilities across their range, Uniform Distribution has a constant probability
    5 min read
    Binomial Distribution in Data Science
    Binomial Distribution is used to calculate the probability of a specific number of successes in a fixed number of independent trials where each trial results in one of two outcomes: success or failure. It is used in various fields such as quality control, election predictions and medical tests to ma
    7 min read
    Poisson Distribution in Data Science
    Poisson Distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space given a constant average rate of occurrence. Unlike the Binomial Distribution which is used when the number of trials is fixed, the Poisson Distribution is used
    7 min read
    Uniform Distribution | Formula, Definition and Examples
    A Uniform Distribution is a type of probability distribution in which every outcome in a given range is equally likely to occur. That means there is no bias—no outcome is more likely than another within the specified set.It is also known as rectangular distribution (continuous uniform distribution).
    11 min read
    Exponential Distribution
    The Exponential Distribution is one of the most commonly used probability distributions in statistics and data science. It is widely used to model the time or space between events in a Poisson process. In simple terms, it describes how long you have to wait before something happens, like a bus arriv
    3 min read
    Normal Distribution in Data Science
    Normal Distribution also known as the Gaussian Distribution or Bell-shaped Distribution is one of the widely used probability distributions in statistics. It plays an important role in probability theory and statistics basically in the Central Limit Theorem (CLT). It is characterized by its bell-sha
    6 min read
    Mathematics | Beta Distribution Model
    The Beta Distribution is a continuous probability distribution defined on the interval [0, 1], widely used in statistics and various fields for modeling random variables that represent proportions or probabilities. It is particularly useful when dealing with scenarios where the outcomes are bounded
    11 min read
    Gamma Distribution Model in Mathematics
    Introduction : Suppose an event can occur several times within a given unit of time. When the total number of occurrences of the event is unknown, we can think of it as a random variable. Now, if this random variable X has gamma distribution, then its probability density function is given as follows
    2 min read
    Chi-Square Test for Feature Selection - Mathematical Explanation
    One of the primary tasks involved in any supervised Machine Learning venture is to select the best features from the given dataset to obtain the best results. One way to select these features is the Chi-Square Test. Mathematically, a Chi-Square test is done on two distributions two determine the lev
    4 min read
    Student's t-distribution in Statistics
    As we know normal distribution assumes two important characteristics about the dataset: a large sample size and knowledge of the population standard deviation. However, if we do not meet these two criteria, and we have a small sample size or an unknown population standard deviation, then we use the
    10 min read
    Python - Central Limit Theorem
    Central Limit Theorem (CLT) is a foundational principle in statistics, and implementing it using Python can significantly enhance data analysis capabilities. Statistics is an important part of data science projects. We use statistical tools whenever we want to make any inference about the population
    7 min read
    Limits, Continuity and Differentiability
    Limits, Continuity, and Differentiation are fundamental concepts in calculus. They are essential for analyzing and understanding function behavior and are crucial for solving real-world problems in physics, engineering, and economics.Table of ContentLimitsKey Characteristics of LimitsExample of Limi
    10 min read
    Implicit Differentiation
    Implicit Differentiation is the process of differentiation in which we differentiate the implicit function without converting it into an explicit function. For example, we need to find the slope of a circle with an origin at 0 and a radius r. Its equation is given as x2 + y2 = r2. Now, to find the s
    5 min read

    Calculus for Machine Learning

    Partial Derivatives in Engineering Mathematics
    Partial derivatives are a basic concept in multivariable calculus. They convey how a function would change when one of its input variables changes, while keeping all the others constant. This turns out to be particularly useful in fields such as physics, engineering, economics, and computer science,
    10 min read
    Advanced Differentiation
    Derivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. Geometrically speaking, the derivative of any function at a particular point gives the slope of the tangent at that point of
    8 min read
    How to find Gradient of a Function using Python?
    The gradient of a function simply means the rate of change of a function. We will use numdifftools to find Gradient of a function. Examples: Input : x^4+x+1 Output :Gradient of x^4+x+1 at x=1 is 4.99 Input :(1-x)^2+(y-x^2)^2 Output :Gradient of (1-x^2)+(y-x^2)^2 at (1, 2) is [-4. 2.] Approach: For S
    2 min read
    Optimization techniques for Gradient Descent
    Gradient Descent is a widely used optimization algorithm for machine learning models. However, there are several optimization techniques that can be used to improve the performance of Gradient Descent. Here are some of the most popular optimization techniques for Gradient Descent: Learning Rate Sche
    4 min read
    Higher Order Derivatives
    Higher order derivatives refer to the derivatives of a function that are obtained by repeatedly differentiating the original function.The first derivative of a function, f′(x), represents the rate of change or slope of the function at a point.The second derivative, f′′(x), is the derivative of the f
    6 min read
    Taylor Series
    A Taylor series represents a function as an infinite sum of terms, calculated from the values of its derivatives at a single point.Taylor series is a powerful mathematical tool used to approximate complex functions with an infinite sum of terms derived from the function's derivatives at a single poi
    8 min read
    Application of Derivative - Maxima and Minima
    Derivatives have many applications, like finding rate of change, approximation, maxima/minima and tangent. In this section, we focus on their use in finding maxima and minima.Note: If f(x) is a continuous function, then for every continuous function on a closed interval has a maximum and a minimum v
    6 min read
    Absolute Minima and Maxima
    Absolute Maxima and Minima are the maximum and minimum values of the function defined on a fixed interval. A function in general can have high values or low values as we move along the function. The maximum value of the function in any interval is called the maxima and the minimum value of the funct
    11 min read
    Optimization for Data Science
    From a mathematical foundation viewpoint, it can be said that the three pillars for data science that we need to understand quite well are Linear Algebra, Statistics and the third pillar is Optimization which is used pretty much in all data science algorithms. And to understand the optimization conc
    5 min read
    Unconstrained Multivariate Optimization
    Wikipedia defines optimization as a problem where you maximize or minimize a real function by systematically choosing input values from an allowed set and computing the value of the function. That means when we talk about optimization we are always interested in finding the best solution. So, let sa
    4 min read
    Lagrange Multipliers | Definition and Examples
    In mathematics, a Lagrange multiplier is a potent tool for optimization problems and is applied especially in the cases of constraints. Named after the Italian-French mathematician Joseph-Louis Lagrange, the method provides a strategy to find maximum or minimum values of a function along one or more
    8 min read
    Lagrange's Interpolation
    What is Interpolation? Interpolation is a method of finding new data points within the range of a discrete set of known data points (Source Wiki). In other words interpolation is the technique to estimate the value of a mathematical function, for any intermediate value of the independent variable. F
    7 min read
    Linear Regression in Machine learning
    Linear regression is a type of supervised machine-learning algorithm that learns from the labelled datasets and maps the data points with most optimized linear functions which can be used for prediction on new datasets. It assumes that there is a linear relationship between the input and output, mea
    15+ min read
    Ordinary Least Squares (OLS) using statsmodels
    Ordinary Least Squares (OLS) is a widely used statistical method for estimating the parameters of a linear regression model. It minimizes the sum of squared residuals between observed and predicted values. In this article we will learn how to implement Ordinary Least Squares (OLS) regression using P
    3 min read

    Regression in Machine Learning

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