Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

Survival

Energy

Light

Heat

Sound

Music

Language

Stories

Work

States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

Electron Configurations

Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

Growth

Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

Behaviors

Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

Algorithms

Tools

Networks

Search

Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

The Normal Probability Curve

(a) Suppose that the frequencies of some data is normally distributed and figure 118.5 represents the probability curve. What is the probability of a value occurring between *a* and *b*?

(b) The weight of a large number of grapefruits were found to be normally distributed with a mean of 1 lb and a standard deviation of 3 oz. What is the probability that any one grapefruit has a weight between 1 lb 3 oz and 1 lb 6 oz?

(c) The average number of persons joining a certain queue in one minute is 2. What is the probability that 5 persons will join the queue in one minute?

**The strings**:
S_{7}P_{6}A_{64} (Grouping - Multi-criteria).
**The math**:

Pj Problem of Interest is of type *grouping* (multi-criteria). *Grouping* is at the heart of statistics. The grouping may be permutational or combinational, single criterion or multi-criteria grouping.

(a) The probability that a value lies within a and b is given by the area under the curve between a and b

So, The probability that a value lies within a and b = ∫_{a} ^{b} f(x) dx.

Where f(x) is as indicated in figure 118.5

Mean = arithmetic average = [f_{1}x_{1} + f_{2}x_{2} ... f_{n}x_{n}]/n

Where f_{i} is the number of x_{i} in the data. i = 1, 2,...n.

Standard deviation σ = [Σ_{i} [f_{i}(x_{i} - mean)^{2}]/n]^{1/2}. i = 1, 2,...n.

(b) Normal frequency curves are varied. However, they are all characterized by their mean and standard deviation. Irrespective of the value of the mean and standard deviation, 68.2 % of the data lie within one standard deviation (σ) on either side of the mean; 95.4 % of the data lie within 2σ on either side of the mean and 99.8 % lie within 3σ on either side of the mean.

The interval of interest is between σ and 2σ to the right of the mean.

So, desired probability = 0.954/2 - 0.682/2 = 0.477 - 0.341 = 0.136.

(c) The *Poisson distribution* is the pertinent probability distribution of interest

The *Poisson distribution* says that if the mean number of events of a particular type in a fixed time interval is μ, then
the probability of n events p(n) occurring in one interval is given by:

p(n) = (μ^{n}e^{-μ})/ n! Where n! is n factorial = n(n-1)(n-2)...1

So, p(5) = (2^{-2})/(5x4x3x2x1) = [32/(2.718)^{2}]/120 = 0.036.

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

Single Variable Functions

Conics

Ordinary Differential Equations (ODEs)

Vector Spaces

Real Numbers

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

More Pj Problem Strings