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The mathematical expectation E[X] of a random variable locates the center of mass for the induced distribution, and the expectation of the square of the distance between X and E[X]measures the spread of the distribution about its center of mass. These quantities are also known, respectively, as the mean (moment) of X and the variance or second moment of X about the mean. Other moments give added information. We examine the expectation of certain functions of X. Each of these functions involves a parameter, in a manner that completely determines the distribution. We refer to these as transforms. In particular, we consider three of the most useful of these: the moment generating function, the characteristic function, and the generating function for nonnegative, integer-valued random variables.

As pointed out in the units on Expectation and Variance , the mathematical expectation E [ X ] = μ X of a random variable X locates the center of mass for the induced distribution, and the expectation

E [ g ( X ) ] = E [ ( X - E [ X ] ) 2 ] = Var [ X ] = σ X 2

measures the spread of the distribution about its center of mass. These quantities are also known, respectively, as the mean (moment) of X and the second moment of X about the mean. Other moments give added information. For example, the third moment about the mean E [ ( X - μ X ) 3 ] gives information about the skew, or asymetry, of the distribution about the mean. We investigatefurther along these lines by examining the expectation of certain functions of X . Each of these functions involves a parameter, in a manner that completely determines the distribution.For reasons noted below, we refer to these as transforms . We consider three of the most useful of these.

Three basic transforms

We define each of three transforms, determine some key properties, and use them to study various probability distributions associated with random variables. In the section on integral transforms , we show their relationship to well known integral transforms. These have been studied extensivelyand used in many other applications, which makes it possible to utilize the considerable literature on these transforms.

Definition . The moment generating function M X for random variable X (i.e., for its distribution) is the function

M X ( s ) = E [ e s X ] ( s is a real or complex parameter)

The characteristic function φ X for random variable X is

φ X ( u ) = E [ e i u X ] ( i 2 = - 1 , u is a real parameter)

The generating function g X ( s ) for a nonnegative, integer-valued random variable X is

g X ( s ) = E [ s X ] = k s k P ( X = k )

The generating function E [ s X ] has meaning for more general random variables, but its usefulness is greatest for nonnegative, integer-valued variables, and we limit ourconsideration to that case.

The defining expressions display similarities which show useful relationships. We note two which are particularly useful.

M X ( s ) = E [ e s X ] = E [ ( e s ) X ] = g X ( e s ) and φ X ( u ) = E [ e i u X ] = M X ( i u )

Because of the latter relationship, we ordinarily use the moment generating function insteadof the characteristic function to avoid writing the complex unit i . When desirable, we convert easily by the change of variable.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
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J, combine like terms 7x-4y
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I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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?
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
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
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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