The law od diminishing marginal utility is given by Alfred Marshall . This topic relates the utility in to majorly three forms : Initial utility which is the satisfaction consumer derives with the consumption of any commodity at a given point of time . Secondly Marginal utility which is diminshing , zoro and sometimes negative even . Whenever a consumer consumes more and more units of a single commodity the marginal utility goes on diminshing . Another aspect is total utility which is the sum total of utility which consumer gets while the consumption of any commodity , total utility increases, maximum and starts decreasing .
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Dispersion : Quartile Deviation in Discrete Series
Quartile deviation is also known as the semi-interquartile range, is a measure of statistical dispersion. It indicates the spread of the middle 50% of a dataset. The quartile deviation is calculated using the first quartile (Q1) and the third quartile (Q3). The formula is:
Quartile Deviation=𝑄3−𝑄1/2
Coefficient of Quartile Deviation = 𝑄3−𝑄1/𝑄3+𝑄1
Here’s a step-by-step explanation:
Arrange Data: Organize the data set in ascending order.
Find Quartiles:
Q1 (First Quartile): The median of the lower half of the dataset (not including the median if the dataset has an odd number of observations).
Q3 (Third Quartile): The median of the upper half of the dataset (not including the median if the dataset has an odd number of observations).
Calculate Quartile Deviation: Subtract Q1 from Q3 and divide by 2.
The quartile deviation provides a robust measure of spread as it is not affected by extreme values or utliers. afterwards find coefficient of quartile deviation by formula QD = 𝑄3−𝑄1/𝑄3+𝑄1 you can watch the video for practical solution of this in various type of series like Individual Series , Discrete Series and Continuous Series. Here in this lecture you will find the Practical Solution in Discrete Series , kindly check the link here and do Subscribe to the channel :
Thanks a Lot
jatin
Multiplication of Matrices
Matrix multiplication is a binary operation that produces a new matrix from two matrices. Unlike addition and subtraction, the dimensions of the matrices involved in multiplication determine the possibility and the result of the operation. Matrix Multiplication involves two major conditions practically for finding AB where A assumes first matrix and B as Second matrix.
1.The no. of columns of first matrix should be equal to the no. of rows of second matrix only then multiplication is possible . if they are not equal then multiplication is not possible.
2. Multiply first row of first matrix with first column of second matrix then first row of first matrix with second column of second matrix then first row of first matrix with third column of second matrix then second row of first matrix with first column of second matrix and so on till the no. of row of first matrix and no. of column of second matrix.
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Matrix multiplication is a fundamental operation in linear algebra, widely used in various fields such as computer graphics, physics, economics, and statistics. Understanding its definition, properties, and application is crucial for effectively utilizing matrices in mathematical and applied contexts.
Addition & Subtraction of Matrices
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The numbers in a matrix are called its elements or entries. A matrix with mmm rows and nnn columns is called an m×nm \times nm×n matrix, read as “m by n matrix”.
Addition of Matrices : Matrix addition is a binary operation that takes two matrices of the same dimensions and produces another matrix of the same dimensions, where each element of the resulting matrix is the sum of the corresponding elements of the input matrices.
Subtraction of Matrices : Matrix subtraction is a binary operation that takes two matrices of the same dimensions and produces another matrix of the same dimensions, where each element of the resulting matrix is the difference of the corresponding elements of the input matrices.
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How to Solve Crammer’s Rule of Matrix
Cramer’s rule is a mathematical theorem used to solve a system of linear equations with as many equations as unknowns, provided that the system has a unique solution. It is applicable to systems of linear equations represented in matrix form. The rule is named after Gabriel Cramer, an 18th-century Swiss mathematician.
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Linear Programming Method (LPP)
Linear Programming (LP) is a mathematical method used to optimize a system with linear relationships subject to certain constraints. It’s widely applied in various fields such as economics, engineering, business management, and logistics, to name a few.
Here’s a basic overview of the Linear Programming method:
Objective Function: This is the function you want to maximize or minimize. It’s usually represented as a linear combination of decision variables.
Decision Variables: These are the variables that represent the quantities you’re trying to find. They’re the parameters you can control or decide upon to optimize the objective function.
Constraints: These are the limitations or restrictions within which the decision variables must operate. Constraints are represented as linear inequalities or equalities.
The steps to solve a Linear Programming problem are as follows:
Formulate the Objective Function: Clearly define what you want to optimize. This could be maximizing profit, minimizing cost, maximizing production, etc.
Identify Decision Variables: Determine the variables that affect the objective function.
Establish Constraints: Identify the limitations on the decision variables. Constraints could be capacity limits, resource availability, demand requirements, etc.
Graphical Method (Optional): For problems with two decision variables, you can visualize the feasible region and optimize the objective function graphically.
Use Linear Programming Software or Algorithms: For problems with more than two decision variables or complex constraints, linear programming software like MATLAB, Python’s PuLP library, or commercial solvers such as CPLEX and Gurobi are used.
Solve the LP Problem: The LP solver finds the optimal solution by iteratively adjusting the decision variables within the constraints to maximize or minimize the objective function.
Interpret the Results: Once the optimal solution is obtained, interpret the results in the context of the problem. This includes understanding the values of decision variables and the optimized value of the objective function.
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Linear Programming is a powerful tool for optimization and decision-making in various real-world scenarios due to its simplicity and efficiency.