Sunday, June 28, 2009

Mathematical Induction - An overview

Mathematical induction is a method of mathematical proof to used to establish that a given statement is true of all natural numbers. It consist of two step , I- Basis step and II- Induction Step. In First step we check the statement is true for n= 1 and when it is true them we proceed to step no-2 , induction step where we have to prove that given statement is true for n+1.
The simplest and most common form of mathematical induction proves that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
The basis (base case): showing that the statement holds when n = 0 or n = 1.
The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.
The assumption in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.
Suppose the sum of first n natural number is given by : 1+2+3+….+n = n(n+1)/2

In this statement we have to prove first this statement is true for n=1 and then n= n+1 in induction step.

Wednesday, June 24, 2009

What is cartesian product in Set Theory ?

In Set Theory, a cartesian product is the direct product of two sets call X and Y. It is denoted by X*Y , is the set of all possible ordered pairs whose first element is a member of X and Second element is from the Y.
A Cartesian product of two finite sets is represented by a table, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table, by choosing the element of the set from the row and the column.
The cartesian product is named after Rene Descartes.
For example - two coins are tossed whose possibilities are Head ( H) and Tail ( T) then the cartesian product of X and Y is
X*Y = ( H,T) , ( H,H) , ( T, H) , ( T,T)
Keep reading

Wednesday, June 17, 2009

Poisson distribution - A discrete Pobability distribution

Simeon Desnis Possion was the person who discovered this distribution. This is also pronounced as "Pwason". This probability distribution expresses the probability of a no. of events occuring in a fixed period of time if these events occurs with a average rate and independently of the time since the last event happened.
This distribution is mainly used for the number of events in a specified intervals such as distance , area or volume.
The fomula given my Possion is as :
f ( k ; y) = y power k x e power -y / K !
where -
e is the base of natural logarthim (e = 2.71828...)
k is the number of occurrences of an event - the probability of which is given by the function
k! is the factorial of k
y is a positive real number , equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average 4 times per minute , and you are interested in the number of events occurring in a 10 minute interval, you would use as your model a Poisson distribution with y = 10×4 = 40.
The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. A classic example is the nuclear decay of atoms or the no. of accidents happened due to the fallen from roof in a metro city.

Tuesday, June 16, 2009

Binomial Theorem.. the expansion of a series

In Mathematics, the binomial theorem is an important formula giving the expansion of powers of sums and its simplest formula is as follows -
For any value of n, whether positive, negative, integer or non-integer, the value of the nth power of a binomial is given by:
(a+b) power n = a power n + na power n-1xb + n(n-1)/2 a power n-2 x b power 2 +.......+ b power n
By this formula , any series can be expanded.
This formula is given by the Blaise PAscal who described that in 17th century but The work on the formula had been started before Pascal gave it to the world.
Now try to expand the (x+y) power 10 by this formula.

Wednesday, June 10, 2009

Creativity ...Separates the common man

Do you know ?
The Bell was invented by an Astronomer - Edmund Halley.
The Pneumatic tyre was invented by a veterinarian - John Dunlop
The safety razor was invented by a sales man - KC Gillete
The vacumm cleaner was invented by a bridge builder - Hubert Booth
There are other many example of the people who think out of box and gave the world new things. How? Because they were the creative people.

Who are the creative prople? Some characterstics are as follows-
•-Optimistic about future
•-Open to alternatives
•-Day dreamers
•-Highly curious and observant
•-Independent thinkers
•-Good at turning innovative ideas into practical solutions
•-Take action and make things happen
•-Adventurous with multiple interest

Monday, June 8, 2009

What is Regression Analysis

How we establish a relationship with a dependent variable and Independent variable? The answer is in Statistics in the form of Regression Analysis. This is a technique for the analysis of numerical data consisting of two variable - Dependent variable also known as Response Variable and Independent Variable also known as Explantory variable. Most commonly the best fit is evalauted and calculated by the Least squares method.
What is the use of Regression Analysis ? Regression is used for forecasting , inference, hypothesis testing and casual relationships.
The term " Regression " was given by Francis Galton.
It is convenient to assume an environment in which an experiment is performed: the dependent variable is then outcome of a measurement.
Y= F ( X, B)
The regression equation deals with the following variables:
The unknown parameters denoted as β; this may be a scalar or a vector of length k.
The independent variables, X.
The dependent variable, Y.
Regression equation is a function of variables X and β.
and the equation of straight line is
y= ax+b where we have to find out the value of a and b then we find out the relation between the independent variable and dependent variable.
Keep reading...........

Thursday, June 4, 2009

Admission time- What are the things to keep in mind to enrol in any course?

This is admission time once again and lacs of students are enrolling themselves to various courses. Followings are the tips to all the aspirants to enrol in any course

1. Idnetify your inborn assets - Idnetify your talent and select the course of your choice. Don't flow in the river without knowing its direction.

2. Look the prospective of the course and demand in future of the course for which you want to go for.

3. Try to enrol in a course which you feel you can do better and channelise your energy.

4. Career making is big decision to every individual , so, If you select a course, you judge yourselves that this course suits you or not. Asked yourself and visualise yourself.

5. Do your SWOT analysis ( Strength, Weakness, Opportuntiies and Threats).

Do take care of these points.

Wednesday, June 3, 2009

What is Correlation analysis?

Assume one situtation, when we add the sugar into tea , this increases the sweetness of tea. More sugar , means more sweetness of tea. Here tea and sugar are two variables whcih goes into one direction. There is a correlation between these two variables. This is the correlation and Correlation is a measure of association between two variables.
The value of a correlation coefficient ranges from -1 to +1. A minus one indicates a perfect negative correlation, while a plus one indicates a perfect positive correlation. A correlation of zero means there is no relationship between the two variables. When there is a negative correlation between two variables, as the value of one variable increases, the value of the other variable decreases, and vise versa. In other words, for a negative correlation, the variables work opposite each other. When there is a positive correlation between two variables, as the value of one variable increases, the value of the other variable also increases. The variables move together. The two most popular correlation coefficients are: Spearman's correlation coefficient rho and Pearson's product-moment correlation coefficient.
Visit regularly, there is lot more to come............

Monday, June 1, 2009

Role of Quantitative Techniques in Management

I am a visiting faculty in a management institute where I do teach Quantitative Techniques to students who are enrolled for MBA programmes. On last sunday, When I was teaching them, one students asked me about the role of Quantitative Techniques in Management. I have explained them why QT is important in management syallbus.

Quantitative approach to management give a clear cut picture to management what is going on , what was the past trend and what can be the future in quantitaive terms. QT approach give so many statistical tools to help management in decision making. Matices, Measure of central tendency, Probability, Transportation and assignment problem, Corelation and regression are some of the tools which helps a lot in decision making.

So, if you want to sharpen your management skills , you have to learn the quantitative methods.