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This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by EwaPaszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant.

Normal distribution

The normal distribution is perhaps the most important distribution in statistical applications since many measurements have (approximate) normal distributions. One explanation of this fact is the role of the normal distribution in the Central Theorem.

Briefly, we say that X is N ( μ , σ 2 )

Proof of the p.d.f. properties

Clearly, f ( x ) > 0 . Let now evaluate the integral: I = 1 σ 2 π exp [ ( x μ ) 2 2 σ 2 ] d x , showing that it is equal to 1. In the integral, change the variables of integration by letting z = ( x μ ) / σ . Then,

I = 1 2 π e z 2 / 2 d z , since I > 0 , if I 2 = 1 , then I = 1 .

Now I 2 = 1 2 π [ e x 2 / 2 d x ] [ e y 2 / 2 d y ] , or equivalently,

I 2 = 1 2 π exp ( x 2 + y 2 2 ) d x d y .

Letting x = r cos θ , y = r sin θ (i.e., using polar coordinates), we have

I 2 = 1 2 π 0 2 π 0 e r 2 / 2 r d r d θ = 1 2 π 0 2 π d θ = 1 2 π 2 π = 1.

The mean and the variance of the normal distribution is as follows:

E ( X ) = μ and V a r ( X ) = μ 2 + σ 2 μ 2 = σ 2 .

That is, the parameters μ and σ 2 in the p.d.f. are the mean and the variance of X .

Normal distribution

Probability Density Function
Cumulative Distribution Function
p.d.f. and c.d.f graphs of the Normal Distribution

If the p.d.f. of X is

f ( x ) = 1 32 π exp [ ( x + 7 ) 2 32 ] , < x < , then X is N ( 7,16 )

That is, X has a normal distribution with a mean μ =-7, variance σ 2 =16, and the moment generating function

M ( t ) = exp ( 7 t + 8 t 2 ) .

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Questions & Answers

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RAW Reply
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I think
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Brian Reply
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Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Damian Reply
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Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
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Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
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