Description
Central Limit Theorem: Simulate RVs with Exponential Distribution
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 EE 381 Assignment 04
2 ( ) E( ) ; Var( ) 2 12 X X
a b b a X X µ σ
+ − = = = =
EE 381 Project: Central Limit Theorem
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
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
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
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
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
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
EE 381 Assignment 04
