# 13.1 Convergence and the central limit theorem  (Page 3/4)

 Page 3 / 4

Absolutely continuous examples

By use of the discrete approximation, we may get approximations to the sums of absolutely continuous random variables. The results on discrete variables indicatethat the more values the more quickly the conversion seems to occur. In our next example, we start with a random variable uniform on $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$ .

## Sum of three iid, uniform random variables.

Suppose $X\sim$ uniform $\left(0,\phantom{\rule{0.166667em}{0ex}}1\right)$ . Then $E\left[X\right]=0.5$ and $\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=1/12$ .

tappr Enter matrix [a b]of x-range endpoints [0 1] Enter number of x approximation points 100Enter density as a function of t t<=1 Use row matrices X and PX as in the simple caseEX = 0.5; VX = 1/12;[z,pz] = diidsum(X,PX,3);F = cumsum(pz); FG = gaussian(3*EX,3*VX,z);length(z) ans = 298a = 1:5:296; % Plot every fifth point plot(z(a),F(a),z(a),FG(a),'o')% Plotting details (see [link] )

For the sum of only three random variables, the fit is remarkably good. This is not entirely surprising, since the sum of two gives a symmetric triangulardistribution on $\left(0,\phantom{\rule{0.166667em}{0ex}}2\right)$ . Other distributions may take many more terms to get a good fit. Consider the following example.

## Sum of eight iid random variables

Suppose the density is one on the intervals $\left(-1,-0.5\right)$ and $\left(0.5,1\right)$ . Although the density is symmetric, it has two separate regions of probability. From symmetry, $E\left[X\right]=0$ . Calculations show $\mathrm{Var}\left[X\right]=E\left[{X}^{2}\right]=7/12$ . The MATLAB computations are:

tappr Enter matrix [a b]of x-range endpoints [-1 1] Enter number of x approximation points 200Enter density as a function of t (t<=-0.5)|(t>=0.5) Use row matrices X and PX as in the simple case[z,pz] = diidsum(X,PX,8);VX = 7/12; F = cumsum(pz);FG = gaussian(0,8*VX,z); plot(z,F,z,FG)% Plottting details (see [link] )

Although the sum of eight random variables is used, the fit to the gaussian is not as good as that for the sum of three in Example 4 . In either case, the convergence is remarkable fast—only a few terms are needed for good approximation.

## Convergence phenomena in probability theory

The central limit theorem exhibits one of several kinds of convergence important in probability theory, namely convergence in distribution (sometimes called weak convergence). The increasing concentration of values of the sample average random variable A n with increasing n illustrates convergence in probability . The convergence of the sample average is a form of the so-called weak law of large numbers . For large enough n the probability that A n lies within a given distance of the population mean can be made as near one as desired. The fact that the variance of A n becomes small for large n illustrates convergence in the mean (of order 2).

$E\left[|{A}_{n}-\mu {|}^{2}\right]\to 0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{as}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\to \infty$

In the calculus, we deal with sequences of numbers. If $\left\{{a}_{n}:1\le n\right\}$ is a sequence of real numbers, we say the sequence converges iff for N sufficiently large a n approximates arbitrarily closely some number L for all $n\ge N$ . This unique number L is called the limit of the sequence. Convergent sequences are characterized by the fact that for largeenough N , the distance $|{a}_{n}-{a}_{m}|$ between any two terms is arbitrarily small for all $n,\phantom{\rule{0.277778em}{0ex}}m\ge N$ . Such a sequence is said to be fundamental (or Cauchy ). To be precise, if we let $ϵ>0$ be the error of approximation, then the sequence is

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
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
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
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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