Download Approximating the Cumulative Distribution and the Poisson

Approximating the Cumulative Distribution and the Poisson Distribution
by the Normal Distribution
Binomial Distribution:
The binomial distribution represents the total number of
successes out of n Bernoulli trials where only two outcomes
are possible, the probability of success for each trial is
constant, and all trials are independent of each other .
By: Tamara Bender
Project: Prove that the cumulative binomial distributions and the
Poisson distributions can be approximated by the Normal distribution
and that that approximation gets better as the numbers increase.
Normal Approximation to Cumulative Binomial
Distribution:
Proof of the Central Limit Theorem:
Cumulative Binomial Distribution:
The cumulative binomial distribution refers to a specific
range of data being collected in a binomial distribution.
Both the cumulative binomial distribution and the Poisson
distribution can be approximated by the normal distribution by
using the central limit theorem.
To find the moment generating function for the normal
distribution, first begin by letting X be a normal random
variable with a mean µ and a standard deviation σ.
First define a random variable of Zi by
.We can
determine that the mean of Zi will be zero and the variance
will be one. The moment generating function for Zi is
The Poisson Distribution:
The Poisson distribution represents the probability of a
number of times a random event occurs in a given amount
of time unit.
1
M X t  
 2
1

 2
The Normal Distribution:
The normal distribution is a probability distribution that is
used to approximate continuous random variables around a
single mean value.
tx 



tx
e e

1  x  
2 2
2
dx
From the stated central limit theorem, we know that
2

1  x   
exp
tx


  2  2  dx



2
2
2
4 2
1  x  
1  x    t    2 t   t


2 2
2
2
2
x



t

  t  1 2t 2
1

2
2
2
1

 1
M X  t   exp  t   2t 2 
2

  2



By the defined properties of moment generating functions,
the sum of independent random variables is the product of
individual moment generating functions.
 1  x    t 2  
dx
exp  
 2

2


Moment Generating Functions:
1

 =
 exp  t  2t 2 
Moment generating functions redefines a specific
2


probability distribution by using expected values of a
random variable. The moment generating function of a
random variable X is the function MX(t)= E(etx). The
function can then be rewritten since the term etx can be
The Central Limit Theorem:
approximated around zero using a Taylor series expansion. The central limit theorem states that given a distribution
Thus the moment function becomes:
with a mean and a variance, the sampling distribution of the
mean approaches a normal distribution as the same size
As n∞ all the terms but the first term will go to zero. For the
1 2 t0
1 3 t0
2
3
increases. The CLT assumes Y1, Y2,…,Yn are independent


tx
0
t0
M X  t   E e   E e  te  x  0  t e  x  0  t e  x  0 
remaining term we must take the limit of mn(t) as n∞
2
6

 variables that are distributed identically with a mean µ and a
2
3
finite
variance
.
It
then
defines
the
random
variable
U
as
n
t
t

 1 E  x t  E  x2   E  x3  
the following:
2
6
The normal distribution can be written as a moment
generating function, which will be used in the proof of
the central limit theorem.
Normal Approximation to Poisson Distribution:
Recall the moment generating function for the normal distribution
being
. Since we stated that the mean is zero and the
variance is one we have
. This means that any given random
variable distribution converges towards the normal distribution as n
goes towards ∞. In addition, the approximation becomes more
accurate as the larger n becomes.
Works Cited:
Bain, Lee and Max Engelhardt. Introduction to Probability and Mathematical Statistics. Boston: PWS
Publishers, 1987.
Normal Approximation to a Binomial Random Variable. 2007. 9 April 2011
<http://demonstrations.wolfram.com/NormalApproximationToABinomialRandomVariable/>.
Normal Approximation to a Poisson Random Variable. 2007. 9 April 2011
<http://demonstrations.wolfram.com/NormalApproximationToAPoissonRandomVariable/>.
Cryer, Jon and Jeff Whitmer. Introduction to the Central Limit Theorem. 26 June 1999. 15 April 2011
<http://courses.ncssm.edu/math/Stat_Inst/PDFS/SEC_4_f.pdf>.
Devore, Jay. Probability and Statistics for Engineering and the Sciences. Ed. Jennifer Burger. 4th Edition.
Belmont: Wadsworth Publishing Company, 1995.
Falmagne, Jean-Claude. Lectures in Elementary Probability Theory and Stochastic Processes. New York:
William Barter, 2003.
Lane, David. Central Limit Theorem. 2008. 14 April 2011
<http://davidmlane.com/hyperstat/A14043.html>.
Weisstein, Eric W. "Central Limit Theorem." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/CentralLimitTheorem.html
Weisstein, Eric W. "Normal Distribution." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/NormalDistribution.html
Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/PoissonDistribution.html