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Category: EE 381

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0. Introduction and Background Material

0.1. Simulate a R.V. with Uniform Probability Distribution

The Python function “numpy.random.uniform(a,b,n)” will generate n random

numbers with uniform probability distribution in the open interval [a b, ) . The PDF

of a random variable uniformly distributed in [a b, ) is defined as following:

1 , ( ) ( )

0, otherwise

axb

f x b a

≤ ≤ = −

; and

0,

( ) ( ) () , ( )

1,

x a

x a PX x Fx a x b

b a

x b

<

− ≤ = = ≤ < −

≥

It is noted that the mean and variance of a uniformly distributed random variable X are

given by:

2

2 ( ) E( ) ; Var( ) 2 12 X X

a b b a X X µ σ

+ − = = = =

EE 381 Project: Central Limit Theorem Dr. Chassiakos — Page 2

0.2. Simulate a R.V. with Exponential Probability Distribution

The Python function “numpy.random.exponential(a,n)” will generate n

random numbers with exponential probability distribution.

The PDF of a random variable exponentially distributed is defined as following:

1 1 exp( ), 0 (; )

0, 0

T

t t

f t

t

β β β

− ≥ =

<

From the above definition, the CDF of T is found as:

0, 0

( ) () 1 1 exp( ), 0

t

PT t Ft

t t

β

< ≤= = −− ≥

It is noted that the mean and variance of the exponentially distributed random variable

T are given by:

2 2 E( ) ; Var( ) T T = = µβ σβ T T = =

0.3. Central Limit Theorem

If 1 2 , , XX X n are independent random variables having the same probability

distribution with mean µ and standard deviation σ , consider the sum

n n 1 2 S XX X =++ .

This sum n S is a random variable with mean n S µ µ = n and standard deviation

n S σ σ= n .

The Central Limit Theorem states that as n → ∞ the probability distribution of the

R.V. n S will approach a normal distribution with mean n µS and standard deviation

n σ S , regardless of the original distribution of the R.V. 1 2 , , XX X n . The PDF of the

normally distributed R.V. n S is given by:

2

2

1 ( ) ( ) exp{ } 2 2

n

n n

S

n

S S

x

f s µ

σ π σ

− = −

EE 381 Project: Central Limit Theorem Dr. Chassiakos — Page 3

PROBLEMS

1. The Central Limit Theorem

Central Limit Theorem.

Consider a collection of books, each of which has thickness W . The thickness W is a

RV, uniformly distributed between 1 and 3 cm. The mean and standard deviation of

the thickness will be:

2 1 3 (3 1) 2 2; 0.33; 0.57 2 12 µσ σ ww w

+ − = = = = = .

These books are piled in stacks of n =1,5,10, or 15 books. The width n S of a stack of

n books is a RV (the sum of the widths of the n books). This RV has a mean

n S w µ µ = n and a standard deviation of n S w σ σ= n .

Perform the following simulation experiments, and plot the results.

a) Make n =1and run N =10,000 experiments, simulating the RV 1 S W= .

b) After the N experiments are completed, create and plot a probability histogram

of the RV S

c) On the same figure, plot the normal distribution probability function and

compare the probability histogram with the plot of f x( )

2

2

1 ( ) ( ) exp{ } 2 2

S

S S

x f x µ

σ π σ

− = −

d) Make n = 5 and repeat steps (a)-(c)

e) Make n =10 and repeat steps (a)-(c)

f) Make n =15 and repeat steps (a)-(c)

SUBMIT a report following the guidelines as described in the syllabus.

The report should include, among the other requirements:

• The four histograms for n = {1,5,10,15} and the overlapping normal

probability distribution plots.

• The Python code, included in an Appendix.

• Make sure that the graphs are properly labeled.

An example of the graph for n = 2 is shown below.

The code below provides a suggestion on how to generate a bar graph for a

continuous random variable X , representing the total bookwidth, when

n = 2 . Note that X has already been calculated.

The code shows the bar graph plotting only. It does not show the calculations

for X s and it does not show the plotting of the Gaussian function.

EE 381 Project: Central Limit Theorem Dr. Chassiakos — Page 4

Note that the value of ”barwidth” is adjusted as the number of bins

changes, to provide a clear and understandable bar graph.

Also note that the ”density=True” parameter is needed to ensure that the

total area of the bargraph is equal to 1.0.

# X is the array with the values of the RV to be plotted

a=1; b=3; # a=min bookwidth ; b=max bookwidth

nbooks=2; nbins=30; # Number of books ; Number of bins

edgecolor=’w’; # Color separating bars in the bargraph

#

# Create bins and histogram

bins=[float(x) for x in linspace(nbooks*a, nbooks*b,nbins+1)]

h1, bin_edges = histogram(X,bins,density=True)

# Define points on the horizontal axis

be1=bin_edges[0:size(bin_edges)-1]

be2=bin_edges[1:size(bin_edges)]

b1=(be1+be2)/2

barwidth=b1[1]-b1[0] # Width of bars in the bargraph

close(‘all’)

#

fig1=plt.figure(1)

plt.bar(b1,h1, width=barwidth, edgecolor=edgecolor)

EE 381 Project: Central Limit Theorem Dr. Chassiakos — Page 5

2. Exponentially Distributed Random Variables

Exponentially Distributed RVs

The goal is to simulate an exponentially distributed R.V. (T ), given by the

following PDF:

2exp( 2 ), 0 ( ) 0, 0 T

t t

f t t

− ≥ =

<

1. Perform N =10,000 experiments and generate the probability histogram of the

random variable T . Plot the histogram of the RV T .

2. On the same graph, plot the function 2exp( 2 ), 0 ( ) 0, 0

x x

g x

x

− ≥ =

<

and compare to the experimentally generated histogram.

SUBMIT a report following the guidelines as described in the syllabus. The

report should include, among the other requirements:

• the histogram of the RV T ;

• the graph of the function g x( ) overlaying the histogram on the same plot;

• the Python code.

3. Make sure that the graph is properly labeled.

EE 381 Project: Central Limit Theorem Dr. Chassiakos — Page 6

3. Distribution of the Sum of RVs

This problem involves a battery-operated critical medical monitor. The lifetime (T )

of the battery is a random variable with an exponentially distributed lifetime. A

battery lasts an average of τ = 45days. Under these conditions, the PDF of the

battery lifetime is given by:

1 1 exp( ), 0 ( ; ) where 45

0, 0

T

t t

f t

t

β β β β

− ≥ = =

<

As mentioned before, the mean and variance of the random variable T are:

2 2 ; µβ σβ T T = =

When a battery fails it is replaced immediately by a new one. Batteries are

purchased in a carton of 24. The objective is to simulate the RV representing the

lifetime of a carton of 24 batteries, and create a histogram. To do this, follow the

steps below.

a) Create a vector of 24 elements that represents a carton. Each one of the 24

elements in the vector is an exponentially distributed random variable (T ) as

shown above, with β = 45. Use the same procedure as in the previous problem

to generate the exponentially distributed random variable T .

b) The sum of the elements of this vector is a random variable (C ), representing

the life of the carton, i.e.

CTT T =++ 1 2 24

where each , 1,2, 24 T j j = is an exponentially distributed R.V. Create the R.V.

C , i.e. simulate one carton of batteries. This is considered one experiment.

c) Repeat this experiment for a total of N=10,000 times, i.e. for N cartons. Use the

values from the N=10,000 experiments to create the experimental PDF of the

lifetime of a carton, f c( ).

d) According to the Central Limit Theorem the PDF for one carton of 24 batteries

can be approximated by a normal distribution with mean and standard deviation

given by:

24 24 ; 24 24 µ µ β σσ β C T = = C T = =

Plot the graph of a normal distribution with

mean = µC and (standard deviation) = σ C ,

over plot of the experimental PDF on the same figure, and compare the results.

e) Create and plot the CDF of the lifetime of a carton, F c( ) . To do this use the

Python “numpy.cumsum” function on the values you calculated for the

experimental PDF.

EE 381 Project: Central Limit Theorem Dr. Chassiakos — Page 7

Answer the following questions:

1. Find the probability that the carton will last longer than three years, i.e.

P S( 3 365) 1 ( 3 365) 1 (1095) >× =− ≤× =− P S F . Use the graph of the CDF F t( ) to

estimate this probability.

2. Find the probability that the carton will last between 2.0 and 2.5 years (i.e.

between 730 and 912 days): PS F F (730 912) (912) (730) ≤≤ = − . Use the graph of

the CDF F t( ) to estimate this probability.

3. SUBMIT a report following the guidelines as described in the syllabus.

The report should include, among the other requirements:

• The numerical answers using the table below. Note: You will need to replicate the

table, in order to provide the answer in your report. Points will be taken off if you do not use

the table.

• The PDF plot of the lifetime of one carton and the corresponding normal

distribution on the same figure.

• The CFD plot of the lifetime of one carton

• Make sure that the graphs are properly labeled.

• The code in an Appendix.

QUESTION ANS.

1. Probability that the carton will last longer than three years

2. Probability that the carton will last between 2.0 and 2.5 years

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