Applied Mathematics

The product rule for differentiation In our Calculus class for Business, Management and Economics students we use the definition of the derivative of a function using limits: We introduce another method of determining derivatives using differentials which offers an intuitive derivation of powerful differentiation formulas. Given . Here is the “engineering“ way to get the [...]

Here we are concerned with making an inference about a population proportion consisting of two categories(like Success or Failures, Pro-Villar or Anti-Villar) from data obtained in a sample of size . The null hypothesis is that the population proportion . Mean and variance of sample proportion From the binomial distribution, we know that the mean [...]

There are many algorithms for generating combinatorial objects in Python, and here we will write a simple function to generate subsets of n objects. The subsets will be represented as an array of indices. We ignore the empty subset. For example, the subsets of 3 objects of sizes 1, 2 and 3 are: [0] [1] [...]

Introduction Given values of paired data, where

Odeheun.py is an implementation of Heun’s method and is a simple modification of the basic Euler method. It is the simplest predictor-corrector method for solving ODEs. ?Download odeheun.py1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 [...]

Here is odeeuler.py, a simple Python program to illustrate the basic Euler method. This method may not be accurate compared with more sophisticated methods but all students of numerical methods should know how to program the routine in any language and understand its behavior on test ode functions. ?Download odeeuler.py""" File odeuler.py   Synopsis double [...]

In this entry we consider the solution of the linear ODE of the form Introduce the integrating factor . Notice that . Multiply both sides of the equation by . This results in . Now focus on the expanded left hand side: is simply the derivative of ! Therefore . Integrating, we have and solving [...]

We begin to collect various ordinary differential equations of the first order, with their solutions. It may take more than a month to complete this blog entry, but I prefer this over presenting a new blog entry after a long time. Comments from readers are valuable and may lead to corrections and improvements over time. [...]

We wish to show an assertion in The mgf, mean, and variance of the negative binomial distribution. that where is a positive integer and the number of terms in the series is infinite. We start with the listed formula from Wikipedia and Mathworld-Wolfram . Now let a= 1 and x = -x. The formula becomes [...]

1. P(Blue) = 15 /(10+15)= 15/ 25. After the blue marble is drawn, there will be 14 Blue and 15 white marbles in the urn. P(Blue| Blue) = 14/(14 + 15) = 14/29. After the second blue marble is drawn, there will be 13 Blue and 34 White marbles in the urn. P(White| Blue,Blue) = [...]