Math Step by Step

Thursday, October 15, 2015

Detail of Poisson Distribution

Saturday, October 10, 2015

Math Tutoring Videos: Personalized.

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Math Tutoring Videos in the following subjects:

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CALCULUS I

CALCULUS II

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DIFFERENTIAL EQUATIONS

LINEAR ALGEBRA

HISTORY OF MATHEMATICS (PROBLEMS)

ABSTRACT ALGEBRA

OPERATIONS RESEARCH

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email: messenger1964@yahoo.com with details/information needed.

Thursday, August 27, 2015

The Golden Ratio in Us

The Golden Ratio has been known for thousands of years. It is an irrational mathematical constant, approximately 1.61803398874989.

Not so simply put: two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one.

The Golden Ratio shows up in various aspects of nature, architecture and art, and interestingly is the limit of the Fibonacci Sequence, which has been shown to be a natural occurring sequence in the leaves, branches, petals and seeds of various plants.

So the Golden Ratio is in the creation of man, and in the creation of God (nature).

The Golden Ratio not only shows up in the Parthenon,but also in the art of Leonardo Da Vinci, Grapham Sutherland's huge tapestry of Christ the King and even Salvador Dali's Sacrament of the Last Supper.

Why?

What is the common thread between all of the created objects by man that they would have this same ratio? The answer is us. So if we express the golden ratio in our art and architecture, is the golden ratio somehow within us?

Let us look at two pictures of the "Golden Rectangle" (also ratio equivalent to Golden Ratio).

As we look at the two Golden Rectangles, which one is more pleasing, easy on the eye? Most people pick the one that is horizontal.

Why?

The golden ratio is a geometric proportion that is the most aesthetically pleasing to the eye and has been the root of countless mysteries over the centuries. In the previous statement is the answer to our question as to why the one is more pleasing than the other is: “the eye.”

Human beings evolved over millions of years, and one of our most important senses has been our vision. Our eyes were required to scan for predatory dangers in our environment. Since our field of vision, and most of the danger is before us, and not above us, we evolved a method of viewing or scanning objects that is in conjunction with the Golden Rectangle. Adrian Bejan, professor of mechanical engineering at Duke’s Pratt School of Engineering, thinks he knows why the golden ratio pops up everywhere: the eyes scan an image the fastest when it is shaped as a golden-ratio rectangle. “When you look at what so many people have been drawing and building, you see these proportions everywhere,” Bejan said. “It is well known that the eyes take in information more efficiently when they scan side-to-side, as opposed to up and down.” Bejan argues that the world – whether it is a human looking at a painting or a gazelle on the open plain scanning the horizon – is basically oriented on the horizontal. Even our sleep is a scanning or “Rapid Eye Movement” of mostly side to side.

So if we are to look to the Golden Ratio and it's origin, we don't have far to look, for the Golden Ratio is in us. From the evolutionary need to scan for danger, our field of vision has found the Golden Ratio and we have expressed this in Art, Architecture and Music. This most Divine of Ratios is not only in us, but in the plants, animals and universe around us, also reminding us that we are not set apart from our Universe, but very much one with it.

Friday, June 5, 2015

Operations Research: Linear Programming 1 [Matrices 1]

	OPERATIONS RESEARCH
	6/5/2015 10:42:43
#	Linear Programming [Matrix Variables and Constraints]
1	Study of Operations Research Information from Matrices.
	Given the matrix below:
	Cost Table	Dist Center D	Dist Center E	# of Rows	# of Columns
	Plant A	80	215	3	2
	Plant B	100	108
	Plant C	102	68

	The following information can be determined:

	VARIABLES
	Variables = # of data in matrix, or counting data cells.
	So for above matrix we can use the following excel formula
	Variables (=COUNTA(C8:D10))	6

	CONSTRAINTS
	Constraints =the number of rows + the number of columns.
	Constraints (=SUM(F8+G8))	5

Saturday, November 8, 2014

Normal Distribution

Below is my Google Docs Spreadsheet for Normal Distributions:

This sheet you cannot change because I have it locked, but if you would like access so you can use my sheet, and just put in your data and get the answer - email me: messenger1964@yahoo.com

Link to Normal Distribution Page - CLICK HERE!
(Use Math Stat Dist IV tab at bottom)

Normal Distribution
	a	μ	X	σ	Answer Φ(a)
		4.4	2	1.6	-1.5

	b	μ	X	σ	Answer Φ(b)
		10	10.3	0.25	1.2
Formula (p.d.f)
f(x) = 1/(σ√2π)*e^(-(x-μ)^(2)/2σ^(2))	a	⇒	⇒	⇒	0.000000085818477
∀ x ∈ R (-∞, +∞)
Google Sheets Formula =					(=(1/(2PI())EXP(-D3-C3)^(2))/2*E3^(2))

Converting P(X) to P(Z) (raw Z score)					Answer Z
P((X-μ)/σ)				a	-1.5
Example: P(X<10.1)= P((X-μ)/σ)<P(10.1-10/.25)				b	1.2
if u = 10, X = 10.1 and σ = .25


Using Standard Norm Table Formulas:
P{Z<a} = Φ(a)			⇒	⇒	-1.5
P{Z>a} = 1-Φ(a)			⇒	⇒	2.5
P{Z<-a} = 1-Φ(a)			⇒	⇒	2.5
P{Z>-a} = Φ(a)			⇒	⇒	-1.5
P{a<Z<b} = Φ(b)-Φ(a)		a<b	⇒	⇒	2.7
P{-a<Z<b} = Φ(b)+Φ(a)-1		a<b	⇒	⇒	-1.3
P{-b<Z<a} = Φ(b)+Φ(a)-1		a<b	⇒	⇒	-1.3
P{-b<Z<a} = Φ(b)-Φ(a)		a<b	⇒	⇒	2.7


Finding Percentiles of Normal Distributions:		Z	μ	σ	Answer (X)
Z = (X-μ)/σ so X = μ+σ*Z		-0.45	4.4	1.2
X = μ+σ*Z =
X=					3.86
Google Sheets Formula =					(=E29+F29*D29)





E[X] =
E[X]=μ		⇒	⇒	⇒	4.4



E[X^(2)] =
σ^(2)+μ^(2)		⇒	⇒	⇒	21.92
Google Sheets Formula =					(=E3^(2)+C3^(2))

Variance
Var = σ^(2) = σ^(2)		⇒	⇒	⇒	2.56



Standard Deviation
St.Dev = √σ		⇒	⇒	⇒	1.6


E[X^(3)] =
6*θ^(3)		⇒	⇒	⇒	24.576



Third Moment of X about the Mean =
E[(X-E[X])^(3)]=
2*θ^(3)		⇒	⇒	⇒	8.192


Moment Generating Function
M(t) = E[e^(tX)] =
e^[μt+((σ^(2)*(t^(2))/2]