<< Chapter < Page Chapter >> Page >
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 ) .

Got questions? Get instant answers now!

Questions & Answers

so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introduction to statistics' conversation and receive update notifications?