<< Chapter < Page Chapter >> Page >
φ ( t ) = E [ e i t X ] and φ n ( t ) = E [ e i t S n * ] = φ n ( t / σ n )

Using the power series expansion of φ about the origin noted above, we have

φ ( t ) = 1 - σ 2 t 2 2 + β ( t ) where β ( t ) = o ( t 2 ) as t 0

This implies

| φ ( t / σ n ) - ( 1 - t 2 / 2 n ) | = | β ( t / σ n ) | = o ( t 2 / σ 2 n )

so that

n | φ ( t / σ n ) - ( 1 - t 2 / 2 n ) | 0 as n

A standard lemma of analysis ensures

| φ n ( t / σ n ) - ( 1 - t 2 / 2 n ) n | n | φ ( t / σ n ) - ( 1 - t 2 / 2 n ) | 0 as n

It is a well known property of the exponential that

1 - t 2 2 n n e - t 2 / 2 as n

so that

φ ( t / σ n ) e - t 2 / 2 as n for all t

By the convergence theorem on characteristic functions, above, F n ( t ) Φ ( t ) .

The theorem says that the distribution functions for sums of increasing numbers of the X i converge to the normal distribution function, but it does not tell how fast. It is instructive to consider some examples, which are easily worked out with the aid of our m-functions.

Demonstration of the central limit theorem

Discrete examples

We first examine the gaussian approximation in two cases. We take the sum of five iid simple random variables in each case. The first variable has six distinct values; thesecond has only three. The discrete character of the sum is more evident in the second case. Here we use not only the gaussian approximation, but the gaussian approximationshifted one half unit (the so called continuity correction for integer-values random variables). The fit is remarkably good in either case with only five terms.

A principal tool is the m-function diidsum (sum of discrete iid random variables). It uses a designated number of iterations of mgsum.

First random variable

X = [-3.2 -1.05 2.1 4.6 5.3 7.2];PX = 0.1*[2 2 1 3 1 1];EX = X*PX' EX = 1.9900VX = dot(X.^2,PX) - EX^2 VX = 13.0904[x,px] = diidsum(X,PX,5); % Distribution for the sum of 5 iid rvF = cumsum(px); % Distribution function for the sum stairs(x,F) % Stair step plothold on plot(x,gaussian(5*EX,5*VX,x),'-.') % Plot of gaussian distribution function% Plotting details (see [link] )
Figure one is a distribution graph. It is titled, distribution for the sum of five iid random variables. The horizontal axis is labeled, X values, and the vertical axis is labeled PX. The values on the horizontal axis range from -20 in increments of 10 to 40. The values on the vertical axis begin at 0 and increase in increments of 0.2 to 1.4.  There are two captions inside the graph. The first reads, X = [-3.2 -1.05 2.1 4.6 5.3 7.2]. The second reads, PX = 0.1*[2 2 1 3 1 1]. There are two graphs, one, a solid blue line, listed as a sum and the other a dashed and dotted line, listed as gaussian, but they both follow the same path and are nearly indistinguishable as they lay on top of one another. The path begins at the bottom-right corner of the graph. It begins completely flat, but increases in slope at an increasing rate until it is halfway across the graph, at approximately the point (10, 0). At this point, it begins decreasing its positive slope until by the far right side of the graph, approximately the point (35, 1), it has again reduced in slope enough to be a horizontal line. Figure one is a distribution graph. It is titled, distribution for the sum of five iid random variables. The horizontal axis is labeled, X values, and the vertical axis is labeled PX. The values on the horizontal axis range from -20 in increments of 10 to 40. The values on the vertical axis begin at 0 and increase in increments of 0.2 to 1.4.  There are two captions inside the graph. The first reads, X = [-3.2 -1.05 2.1 4.6 5.3 7.2]. The second reads, PX = 0.1*[2 2 1 3 1 1]. There are two graphs, one, a solid blue line, listed as a sum and the other a dashed and dotted line, listed as gaussian, but they both follow the same path and are nearly indistinguishable as they lay on top of one another. The path begins at the bottom-right corner of the graph. It begins completely flat, but increases in slope at an increasing rate until it is halfway across the graph, at approximately the point (10, 0). At this point, it begins decreasing its positive slope until by the far right side of the graph, approximately the point (35, 1), it has again reduced in slope enough to be a horizontal line.
Distribution for the sum of five iid random variables.
Got questions? Get instant answers now!

Second random variable

X = 1:3; PX = [0.3 0.5 0.2]; EX = X*PX'EX = 1.9000 EX2 = X.^2*PX'EX2 = 4.1000 VX = EX2 - EX^2VX = 0.4900 [x,px]= diidsum(X,PX,5); % Distribution for the sum of 5 iid rv F = cumsum(px); % Distribution function for the sumstairs(x,F) % Stair step plot hold onplot(x,gaussian(5*EX,5*VX,x),'-.') % Plot of gaussian distribution function plot(x,gaussian(5*EX,5*VX,x+0.5),'o') % Plot with continuity correction% Plotting details (see [link] )
Figure two is a distribution graph. It is titled, distribution for the sum of five iid random variables. The horizontal axis is labeled, X values, and the vertical axis is labeled PX. The values on the horizontal axis range from 5 in increments of 1 to 15. The values on the vertical axis begin at 0 and increase in increments of 0.2 to 1.2.  There are two captions inside the graph. The first reads, X = [1 2 3]. The second reads, PX = [0.3 0.5 0.2]. There are two graphs, one, a solid blue line, labeled step dbn fn, and the other a dashed and dotted line, labeled gaussian, but they both follow the same path. The step dbn fn is a series of horizontal line segments followed by vertical line segments in varying sizes that fit the shape of the smoother curve, the gaussian curve. A third labeled item is a series of small blue circles that sit at the upper corners of the steps of the solid lined curve, labeled, shifted gaussian. The path begins at the bottom-right corner of the graph. It begins completely flat, but increases in slope at an increasing rate until it is halfway across the graph, at approximately the point (5, 0). At this point, it begins decreasing its positive slope until by the far right side of the graph, approximately the point (14, 1), it has again reduced in slope enough to be a horizontal line. Figure two is a distribution graph. It is titled, distribution for the sum of five iid random variables. The horizontal axis is labeled, X values, and the vertical axis is labeled PX. The values on the horizontal axis range from 5 in increments of 1 to 15. The values on the vertical axis begin at 0 and increase in increments of 0.2 to 1.2.  There are two captions inside the graph. The first reads, X = [1 2 3]. The second reads, PX = [0.3 0.5 0.2]. There are two graphs, one, a solid blue line, labeled step dbn fn, and the other a dashed and dotted line, labeled gaussian, but they both follow the same path. The step dbn fn is a series of horizontal line segments followed by vertical line segments in varying sizes that fit the shape of the smoother curve, the gaussian curve. A third labeled item is a series of small blue circles that sit at the upper corners of the steps of the solid lined curve, labeled, shifted gaussian. The path begins at the bottom-right corner of the graph. It begins completely flat, but increases in slope at an increasing rate until it is halfway across the graph, at approximately the point (5, 0). At this point, it begins decreasing its positive slope until by the far right side of the graph, approximately the point (14, 1), it has again reduced in slope enough to be a horizontal line.
Distribution for the sum of five iid random variables.
Got questions? Get instant answers now!

As another example, we take the sum of twenty one iid simple random variables with integer values. We examine only part of the distribution function where most of theprobability is concentrated. This effectively enlarges the x-scale, so that the nature of the approximation is more readily apparent.

Sum of twenty-one iid random variables

X = [0 1 3 5 6];PX = 0.1*[1 2 3 2 2];EX = dot(X,PX) EX = 3.3000VX = dot(X.^2,PX) - EX^2 VX = 4.2100[x,px] = diidsum(X,PX,21);F = cumsum(px); FG = gaussian(21*EX,21*VX,x);stairs(40:90,F(40:90)) hold onplot(40:90,FG(40:90)) % Plotting details (see [link] )
Figure three is a distribution graph. It is titled, partial distribution for sum of 21 iid random variables. The horizontal axis is labeled, x-values, and the vertical axis is labeled PX. The values on the horizontal axis range in value from 40 to 90 at increments of 5, and the vertical axis ranges from 0 to 1 in increments of .1. There are two labeled equations. The first reads, X = [0 1 3 5 6]. The second reads, PX = 0.1*[1 2 3 2 2]. There are two graphs, one a smooth curve, labeled gaussian dbn, and the other a series of steps closely following the smooth curve, labeled Dbn for sum. Both graphs begin at the point (40, 0) at the bottom-left of the graph. The slope of the smooth curve is flat, and increases until approximately (70, 0.5). At this point, the graph continues increasing, but its slope begins decreasing until at approximately (90, 0.99), the path is again nearly flat. The steps follow the smooth curve along the same path. Figure three is a distribution graph. It is titled, partial distribution for sum of 21 iid random variables. The horizontal axis is labeled, x-values, and the vertical axis is labeled PX. The values on the horizontal axis range in value from 40 to 90 at increments of 5, and the vertical axis ranges from 0 to 1 in increments of .1. There are two labeled equations. The first reads, X = [0 1 3 5 6]. The second reads, PX = 0.1*[1 2 3 2 2]. There are two graphs, one a smooth curve, labeled gaussian dbn, and the other a series of steps closely following the smooth curve, labeled Dbn for sum. Both graphs begin at the point (40, 0) at the bottom-left of the graph. The slope of the smooth curve is flat, and increases until approximately (70, 0.5). At this point, the graph continues increasing, but its slope begins decreasing until at approximately (90, 0.99), the path is again nearly flat. The steps follow the smooth curve along the same path.
Distribution for the sum of twenty one iid random variables.
Got questions? Get instant answers now!

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
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+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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 ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
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
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
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
Good
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied probability' conversation and receive update notifications?

Ask