Scilab Manual for
Probability Theory and Random Processes
by Prof Shital Thakkar
Others
Dharmsinh Desai University1
Solutions provided by
Prof Shital Thakkar
Others
Dharmsinh Desai University
July 12, 2017
1 Funded
by a grant from the National Mission on Education through ICT,
http://spoken-tutorial.org/NMEICT-Intro. This Scilab Manual and Scilab codes
written in it can be downloaded from the ”Migrated Labs” section at the website
http://scilab.in
1
Contents
List of Scilab Solutions
3
1 Write a program to find probability of tossing a coin and
rolling a die through large no. of experimentation.
5
2 To generate Uniform, Gaussian and Exponential distributed
data for given mean and variance.
9
3 Write a program to generate M trials of a random experiment having specific number of outcomes with specified
probabilities.
14
4 To find estimated and true mean of Uniform, Gaussian and
Exponential distributed data.
17
5 To find density and distribution function of a function of
random variable Y = 2X + 1. where X is gaussian R.V.
21
6 Estimate the mean and variance of Y = 2X + 1, where X
is a gaussian random variable.
24
7 Plot Joint density and distribution function of sum of two
Gaussian random variable (Z = X + Y).
26
8 Estimate the mean and variance of a R.V. Z = X+Y. Where
X and Y are also random variables.
30
9 Simulation of Central Limit Theorem.
2
32
List of Experiments
Solution 1.1Calculates probability of sum of tossing two dice . . .
Solution 1.2Finds probability of getting Head when a coin is tossed
Solution 2.1Generation of Uniform Data . . . . . . . . . . . . . . .
Solution 2.2Gaussian Data Generation . . . . . . . . . . . . . . . .
Solution 2.3Exponential Data Generation . . . . . . . . . . . . . .
Solution 3.1Random experiment with outputs in specific range . .
Solution 4.1Comaparison of True and estimated statics of Uniform
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solution 4.2Comaparison of True and estimated statics of Gaussian
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solution 4.3Comaparison of True and estimated statics of Exponential Data . . . . . . . . . . . . . . . . . . . . . . . . .
Solution 5.1Density and Distribution plot generation for one function of Random Variable . . . . . . . . . . . . . . . . .
Solution 6.1Statistics of Function of one random variable . . . . .
Solution 7.1Joint Density and Distribution of Function of two random varible . . . . . . . . . . . . . . . . . . . . . . . .
Solution 8.1Estimation of mean and variance of sum of two random
variable . . . . . . . . . . . . . . . . . . . . . . . . . .
Solution 9.1Summation of two random variable leads to Gaussian
density function . . . . . . . . . . . . . . . . . . . . .
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List of Figures
1.1
Finds probability of getting Head when a coin is tossed . . .
7
2.1
2.2
2.3
Generation of Uniform Data . . . . . . . . . . . . . . . . . .
Gaussian Data Generation . . . . . . . . . . . . . . . . . . .
Exponential Data Generation . . . . . . . . . . . . . . . . .
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5.1
Density and Distribution plot generation for one function of
Random Variable . . . . . . . . . . . . . . . . . . . . . . . .
22
7.1
7.2
9.1
Joint Density and Distribution of Function of two random varible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Joint Density and Distribution of Function of two random varible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summation of two random variable leads to Gaussian density
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
27
27
33
Experiment: 1
Write a program to find
probability of tossing a coin
and rolling a die through large
no. of experimentation.
Scilab code Solution 1.1 Calculates probability of sum of tossing two dice
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// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// Program o f T o s s i n g two d i c e and o b s e r v i n g
P r o b a b i l i t y o f sum o f t h e i r f r o n t f a c e .
// f o r e . g . P r o b a b i l i t y o f sum o f two d i c e = 2 i s
1 / 3 6 a s t h e r e a r e 36 p o s s i b i l i t i e s and sum = 2
can // o c c u r o n l y one c o m b i n a t i o n t h a t i s b o t h
face = 1
clc ;
clear ;
clf ;
N = 10000; // Number o f t i m e s t o s s i n g o f d i e
performed
count = 0; // C o u n t e r f o r c o u n t i n g number o f t i m e s
5
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sum o f d i e
for i = 1: N
y1 = ceil ( rand (1) *6) ; // o u t p u t o f d i e 1
y2 = ceil ( rand (1) *6) ; // o u t p u t o f d i e 2
if (( y1 + y2 ) == 3)
// c h e c k f o r sum o f f r o n t
f a c e o f b o t h d i e i s == 3 ( c h a n g e sum and )
count = count + 1; // i n c r e m e n t t h e c o u n t
v a l u e when sum = 3 o c c u r s
end
prob1 ( i ) = count / i ;
// no . o f t i m e s sum o f
d i e = 3/ t o t a l no . t r i a l s
end
plot ( prob1 )
xlabel ( ’ Number o f T r i a l s ’ ) ;
ylabel ( ’ P r o b a b i l i t y ’ ) ;
title ( ’ P r o b a b i l i t y o f g e t t i n g sum o f d o t s on f a c e s
o f a d i c e t o be 3 ’ ) ;
// A s s i g n m e n t : Program can be c h e c k e d f o r o t h e r
v a l u e s o f sum a t l i n e number 1 0 .
Scilab code Solution 1.2 Finds probability of getting Head when a coin
is tossed
1
2
3
4
5
// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// T h i s program f i n d p r o b a b i l i t y o f g e t t i n g Head when
a coin i s tossed .
6 // P r o b a b i l i t y = 1/2 = 0 . 5 a s t h e r e a r e two p o s s i b l e
outcomes in c o i n t o s s i n g experiment .
6
Figure 1.1: Finds probability of getting Head when a coin is tossed
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clc ;
clear all ;
clf ;
a1 = 1000;
count1 = 0;
for i = 1: a1
x = round ( rand (1) ) ;
// round : t h e e l e m e n t s t o n e a r e s t i n t e g e r
// r a n d : r e t u r n s a pseudorandom , s c a l a r v a l u e
drawn from a u n i f o r m
// d i s t r i b u t i o n on t h e u n i t i n t e r v a l .
if ( x ==1) // HEAD− ’ 1 ’ , c o n d i t i o n t h a t d e t e c t s ’
HEAD’ comes o r n o t
count1 = count1 + 1; // i n c r e m e n t t h e c o u n t
v a l u e when head o c c u r s
end
p ( i ) = count1 / i ; // p r o b a b i l i t y o f head o c c u r i n g a t
ith t r a i l
7
21 end
22 plot (1: a1 , p )
23 // p l o t t h e p r o b . a t i t h
trail ( plots discrete
sequence )
xlabel ( ’ Number o f T r i a l s ’ ) ;
ylabel ( ’ P r o b a b i l i t y ’ ) ;
title ( ’ P r o b a b i l i t y o f g e t t i n g head ’ ) ;
// A s s i g n m e n t :
// 1 .
perform above experiment with n = 100 ,1000 .
// 2 .
Extend t h e a b o v e e x p e r i m e n t t o f i n d
p r o b a b i l i t y o f 3 heads in 4 coin t o s s e s .
30 //
Match t h e r e s u l t t h e o r e t i c a l l y .
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8
Experiment: 2
To generate Uniform, Gaussian
and Exponential distributed
data for given mean and
variance.
Scilab code Solution 2.1 Generation of Uniform Data
//To g e n e r a t e Uniform , G a u s s i a n and E x p o n e n t i a l
d i s t r i b u t e d data .
2 // O p e r a t i n g System : Windows XP o r l a t e r ,
3 // S c i l a b
: 5.3.3
4 //NOTE: EXECUTE ONE BY ONE SEGEMENT
1
5
6 // [ 1 ] Uniform Data G e n e r a t i o n
7 clc ;
8 clear all ;
9 // b = i n p u t ( ’ h i g h e r l i m i t o f u n i f o r m r . v . b = ’ ) //
10
Enter h i g h e r l i m i t o f uniform r . v .
// a = i n p u t ( ’ l o w e r l i m i t o f u n i f o r m r . v . a = ’) //
Enter lower l i m i t o f uniform r . v .
9
Figure 2.1: Generation of Uniform Data
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clf () ;
b =4; // h i g h e r r a n g e o f u n i f o r m d a t a
a =2; // l o w e r r a n g e o f u n i f o r m d a t a
x = a + (b - a ) .* rand (1 ,1000 , ’ u n i f o r m ’ ) ; // u n i f o r m d a t a
g e n e r a t i o n o f l e n g t h 100
histplot (1000 , x ) // h i s t o g r a m o f g e n e r a t e d d a t a
xlabel ( ’ Number o f S a m p l e s ’ ) ;
ylabel ( ’ Range o f V a l u e s o f U n i f o r m l y g e n e r a t e d d a t a ’
);
title ( ’ U n i f o r m l y d i s t r i b u t e d d a t a i n t h e r a n g e o f a
to b ’ );
Scilab code Solution 2.2 Gaussian Data Generation
10
Figure 2.2: Gaussian Data Generation
//To g e n e r a t e Uniform , G a u s s i a n and E x p o n e n t i a l
d i s t r i b u t e d data .
2 // O p e r a t i n g System : Windows XP o r l a t e r ,
3 // S c i l a b
: 5.3.3
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// [ 2 ] E x p o e n e n t i a l d a t a g e n e r a t i o n & Mean and
Variance Calcultaion of Exponential d i s t r i b u t e d
data .
clc ;
clear all ;
clf () ;
// ( i ) E x p o n e n t i a l d a t a g e n e r a t i o n
lambda = 2; // o r lambda = i n p u t ( ’ e n t e r lemda v a l u e
f o r e x p o n e n t i a l r . v . ’ ) // lemda o f e x p o n e n t i a l d a t a
;
X = grand (10000 ,1 , ” exp ” , 1/ lambda ) ;
Xmax = max ( X ) ;
histplot (40 , X , style =2)
x = linspace (0 , max ( Xmax ) ,100) ’;
plot2d (x , lambda * exp ( - lambda * x ) , strf = ” 000 ” , style =5)
legend ([ ” e x p o n e n t i a l random s a m p l e h i s t o g r a m ” ” e x a c t
11
Figure 2.3: Exponential Data Generation
d e n s i t y c u r v e ” ]) ;
17 xlabel ( ’ s a m p l e v a l u e ’ ) ;
18 ylabel ( ’ E x p o n e n t i a l o u t p u t v a l u e s ’ ) ;
19 title ( ’ E x p o n e n t i a l d i s t r i b u t e d d a t a ’ ) ;
Scilab code Solution 2.3 Exponential Data Generation
//To g e n e r a t e Uniform , G a u s s i a n and E x p o n e n t i a l
d i s t r i b u t e d data .
2 // O p e r a t i n g System : Windows XP o r l a t e r ,
3 // S c i l a b
: 5.3.3
1
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// [ 3 ] G a u s s i a n d a t a g e n e r a t i o n
// G a u s s i a n d a t a g e n e r a t i o n
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clc ;
clear all ;
clf () ;
//m = i n p u t ( ’ e n t e r mean v a l u e f o r G a u s s i a n r . v . ’ )
// v a r i = i n p u t ( ’ e n t e r mean v a l u e f o r G a u s s i a n r . v . ’ )
m = 2; // mean v a l u e o f g a u s s i a n d a t a
sd = 1; // s t a n d a r d d e v i a t i o n
vari = sd ^2;
X = grand (100000 ,1 , ” n o r ” , m , sd ) ;
Xmax = max ( X ) ;
clf ()
histplot (40 , X , style =2)
x = linspace ( -10 , max ( Xmax ) ,100) ’;
plot2d (x ,(1/( sqrt (2* %pi * vari ) ) ) * exp ( -0.5*( x - m ) .^2/
vari ) , strf = ” 000 ” , style =5)
xlabel ( ’ s a m p l e v a l u e ’ ) ;
ylabel ( ’ G a u s s i a n o u t p u t v a l u e s ’ ) ;
title ( ’ G a u s s i a n d i s t r i b u t e d d a t a ’ ) ;
legend ([ ” G a u s s i a n random s a m p l e h i s t o g r a m ” ” e x a c t
d e n s i t y c u r v e ” ] ,2) ;
13
Experiment: 3
Write a program to generate M
trials of a random experiment
having specific number of
outcomes with specified
probabilities.
Scilab code Solution 3.1 Random experiment with outputs in specific range
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// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// W r i t e a program t o g e n e r a t e M t r i a l s o f a random
experiment having s p e c i f i c
// number o f o u t c o m e s w i t h s p e c i f i e d p r o b a b i l i t i e s .
// h e r e No . o f t r i a l s = 1 0 0 0 , no . o u t c o m e s ( r v ) = 3
w i t h s p e c i e d p r o b a b i l i t y e n t e r e d by u s e r
clc ;
clear all ;
rand ( ’ s e e d ’ ) // c h e c k
M = 1000; // Number o f t r i a l s o f random e x p e r i m e n t
outcomes = 3; // P o s s i b l e number o f o u t c o m e s o f
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random e x p e r i m e n t
for i = 1: outcomes -1
r ( i ) = input ( ’ e n t e r u p p e r r a n g e o f p r o b a b i l i t y
o f r . v . ( v a l u e s i n t h e 0<r <1) : ’ ) // e n t e r
v a l u e s i n t h e 0<r <1
if r ( i ) >1 then
error ( ’ E n t e r v a l u e s i n t h e r a n g e 0<r <1 ’ )
end
end
x = zeros (M ,1) ;
for i = 1: M
u = rand (1 ,1) ; // random outcome
if u <= r (1) then
x (i ,1) =1; // a s s i g n r v v a l u e = 1 i f u<=r ( 1 )
elseif u > r (1) & u <= r (2)
x (i ,1) =2; // s e c o n d r v v a l u e
else
x (i ,1) =3; // t h i r d r v v a l u e
end
end
count1 =0; count2 =0; count3 =0;
for i =1:1000
if x (i ,1) ==1 then
count1 = count1 + 1;
elseif x (i ,1) == 2
count2 = count2 + 1;
else
count3 = count3 + 1;
end
end
estP1 = count1 / M ; disp ( estP1 ) // e s t i m a t e d p r o b a b i l i t y
o f g e n e r a t e d random v a r i a b l e
estP2 = count2 / M ; disp ( estP2 ) // e s t i m a t e d p r o b a b i l i t y
o f g e n e r a t e d random v a r i a b l e
estP3 = count3 / M ; disp ( estP3 ) // e s t i m a t e d p r o b a b i l i t y
o f g e n e r a t e d random v a r i a b l e
// A s s i g n m e n t :
15
// 1 . Extend t h i s program f o r 4 number o f random
variable
45 // 2 . Extend t h i s program f o r more number o f t r i a l s .
i . e . M = 5000 ,10000 etc .
44
16
Experiment: 4
To find estimated and true
mean of Uniform, Gaussian and
Exponential distributed data.
Scilab code Solution 4.1 Comaparison of True and estimated statics of
Uniform Data
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3
4
5
6
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// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// h e r e we g e n e r a t e u n i f o r m l y d i s t r i b u t e d d a t a
compare i t s s t a t i s t i c s w i t h c a l c u l a t e d u s i n g
equation .
// [ 1 ] Mean and V a r i a n c e C a l c u l t a i o n o f U n i f o r m l y
d i s t r i b u t e d data
clc ;
clear all ;
b =4; // h i g h e r r a n g e o f u n i f o r m d a t a
a =2; // l o w e r r a n g e o f u n i f o r m d a t a
x = a + (b - a ) .* rand (1 ,1000) ; // Uniform d a t a g e n e r a t i o n
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using function
Uni_true_mean = mean ( x )
mprintf ( ’ Uniform True Mean = %f ’ , Uni_true_mean )
Uni_true_var = variance ( x )
mprintf ( ’ \n Uniform True Mean = %f ’ , Uni_true_var )
px = 1/( b - a ) // p d f c a l c u l t a i o n o f u n i f o r m r . v .
m_uniform = integrate ( ’ x ∗ px ’ , ’ x ’ ,a , b ) // mean
c a l c u l t a i o n of uniform r . v .
fsq_uniform = integrate ( ’ ( x ˆ 2 ) ∗ px ’ , ’ x ’ ,a , b ) // mean
square value of uniform r . v .
var_uniform = fsq_uniform - ( m_uniform ) .^2 // v a r i a n c e
of uniform r . v .
mprintf ( ’ \n Uniform C a l c u l a t e d Mean = %f ’ , m_uniform )
mprintf ( ’ \n Uniform C a l c u l a t e d V a r i a n c e = %f ’ ,
var_uniform )
Scilab code Solution 4.2 Comaparison of True and estimated statics of
Gaussian Data
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// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// h e r e we g e n e r a t e G a u s s i a n d i s t r i b u t e d d a t a compare
i t s s t a t i s t i c s with c a l c u l a t e d using equation .
// [ 3 ] Mean & V a r i a n c e c a l c u l a t i o n o f G a u s s i a n Data
clc ;
clear all ;
m = 2; // mean v a l u e o f g a u s s i a n d a t a
sd = 1; // s t a n d a r d d e v i a t i o n
vari = sd ^2;
X = grand (10000 ,1 , ” n o r ” , m , sd ) ; // g a u s s i a n d a t a
generation using function
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Gaus_true_mean = mean ( X )
mprintf ( ’ G a u s s i a n True Mean = %f ’ , Gaus_true_mean )
Gaus_true_var = variance ( X )
mprintf ( ’ \n G a u s s i a n True V a r i a n c e = %f ’ ,
Gaus_true_var )
// Mean and v a r i a n c e c a l c u l t a i o n u s i n g f o r m u l a
gs_mean = integrate ( ’ x ∗ ( 1 / s q r t ( 2 ∗ %pi ∗ v a r i ) ∗ exp ( −(x−m)
ˆ 2 / ( 2 ∗ v a r i ) ) ) ’ , ’ x ’ ,0 ,100)
gs_fsq = integrate ( ’ ( x ˆ 2 ) ∗ ( 1 / s q r t ( 2 ∗ %pi ∗ v a r i ) ∗ exp ( −(x−
m) ˆ 2 / ( 2 ∗ v a r i ) ) ) ’ , ’ x ’ ,0 ,100) ;
gs_var = gs_fsq - gs_mean .^2;
mprintf ( ’ \n G a u s s i a n C a l c u l a t e d Mean = %f ’ , gs_mean )
mprintf ( ’ \n G a u s s i a n C a l c u l a t e d V a r i a n c e = %f ’ ,
gs_var )
Scilab code Solution 4.3 Comaparison of True and estimated statics of
Exponential Data
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3
4
5
6
7
8
9
10
// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// h e r e we g e n e r a t e E x p o n e n t i a l l y d i s t r i b u t e d d a t a
compare i t s s t a t i s t i c s w i t h c a l c u l a t e d u s i n g
equation .
// [ 2 ] Mean and V a r i a n c e C a l c u l a t i o n o f E x p o n e n t i a l
data
clc ;
clear all ;
lambda = 2; // o r lambda = i n p u t ( ’ e n t e r lemda v a l u e
f o r e x p o n e n t i a l r . v . ’ ) // lemda o f e x p o n e n t i a l d a t a
;
X = grand (1000 ,1 , ” exp ” , 1/ lambda ) ; // E x p o n e n t i a l d a t a
generation using function
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Expo_true_mean = mean ( X )
mprintf ( ’ E x p o n e n t i a l True Mean = %f ’ , Expo_true_mean )
Expo_true_var = variance ( X )
mprintf ( ’ \n E x p o n e n t i a l True V a r i a n c e = %f ’ ,
Expo_true_var )
// Mean and v a r i a n c e c a l c u l t a i o n u s i n g f o r m u l a
ex_mean = integrate ( ’ x ∗ ( lambda ∗ exp (−lambda ∗ x ) ) ’ , ’ x ’
,0 ,100)
ex_fsq = integrate ( ’ ( x ˆ 2 ) ∗ ( lambda ∗ exp (−lambda ∗ x ) ) ’ , ’ x ’
,0 ,100) ;
ex_var = ex_fsq - ex_mean .^2
mprintf ( ’ \n E x p o n e n t i a l C a l c u l a t e d Mean = %f ’ ,
ex_mean )
mprintf ( ’ \n E x p o n e n t i a l C a l c u l a t e d V a r i a n c e = %f ’ ,
ex_var )
20
Experiment: 5
To find density and distribution
function of a function of
random variable Y = 2X + 1.
where X is gaussian R.V.
Scilab code Solution 5.1 Density and Distribution plot generation for one
function of Random Variable
1
2
3
4
// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
//To f i n d d e n s i t y ( p d f ) and d i s t r i b u t i o n ( c d f )
f u n c t i o n o f a f u n c t i o n o f random v a r i a b l e
5 //Y = 2X + 1 ( h a v i n g form Y = aX + b ) . where X i s
g a us s i an R.V.
6
7 clc ;
8 clear all ;
9 clf () ;
10 mean_x = 1; // mean v a l u e o f
21
g a u s s i a n data
Figure 5.1: Density and Distribution plot generation for one function of
Random Variable
11 sd_x = 1; // s t a n d a r d d e v i a t i o n
12 vari_x = sd_x .^2;
13 lgd = [];
14 //PDF and CDF o f G a u s s i a n Random V a r i a b l e X
15 x = linspace ( -10 ,10 ,100) ;
16 plot2d (x ,((1/( sqrt (2* %pi * vari_x ) ) ) * exp ( -0.5*( x -
mean_x ) .^2/ vari_x ) ) ,2) ; // p l o t s p d f o f X
17 set ( gca () ,” a u t o c l e a r ” ,” o f f ” )
18 plot2d (x , cdfnor ( ”PQ” ,x , mean_x * ones ( x ) , sd_x * ones ( x ) )
,3) ; // c d f o f g a u s s i a n RV X
19 set ( gca () ,” a u t o c l e a r ” ,” on ” )
20 xlabel ( ’ s a m p l e p o i n t s ’ ) ;
21 ylabel ( ’PDF & CDF ’ ) ;
22 // t i t l e ( ’ d e n s i t y and d i s t r i b u t i o n
23
function for
Gaussian function ’ ) ;
legend ([ ’PDF o f n o r m a l d i s t r i b u t i o n ’ ; ’CDF o f n o r m a l
d i s t r i b u t i o n ’ ] ,2) ;
24
22
25 //PDF and CDF o f Y = aX + b where a = 2 , b = 1
26 a = 2;
27 b = 1;
28 y = a * x + b ; // F u n c t i o n o f One Random V a r i a b l e
29 mean_y = a * mean_x + b ;
30 vari_y =( a * sd_x ) .^2;
31 figure (2 , ” B a c k g r o u n d C o l o r ” ,[1 ,1 ,1]) ;
32 plot2d (y ,((1/( sqrt (2* %pi * vari_y * a .^2) ) ) * exp ( -0.5*( y -
mean_y ) .^2/ vari_y ) ) ,2) ; // p d f o f Y
33 set ( gca () ,” a u t o c l e a r ” ,” o f f ” )
34 plot2d (x , cdfnor ( ”PQ” ,y ,( a * mean_x + b ) * ones ( x ) ,( a * sd_x )
* ones ( x ) ) ,3) ; // c d f o f y
35 set ( gca () ,” a u t o c l e a r ” ,” on ” )
36 xlabel ( ’ s a m p l e p o i n t s ’ ) ;
37 ylabel ( ’PDF & CDF o f Y = 2X + 1 ’ ) ;
38 legend ([ ’PDF o f Y = 2X + 1 ’ ; ’CDF o f Y = 2X + 1 ’ ] ,2) ;
39
40
41 // A s s i g n m e n t :
42 // 1 . P e r f o r m t h e o p e r a t i o n f o r f u n c t i o n Y = 5X + 1 .
43 // 2 . G e n e r a t e p d f and c d f o f n o n l i n e a r f u n c t i o n
b e t w e e n Y and X .
23
Experiment: 6
Estimate the mean and
variance of Y = 2X + 1, where
X is a gaussian random
variable.
Scilab code Solution 6.1 Statistics of Function of one random variable
1
2
3
4
5
6
7
8
9
10
11
12
13
14
// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// True and E s t i m a t e d v a l u e o f mean and v a r i a n c e o f
f u n c t i o n o f one random
// v a r i a b l e h a v i n g form o f Y = aX +b .
clc ;
clear all ;
rand ( ’ s e e d ’ , getdate ( ’ s ’ ) )
m = 0; // mean o f random v a r i a b l e x
vari = 1; // v a r i a n c e o f random v a r i a b l e x
m_est = 0;
var_est = 0;
for i = 1:1000
y (i ,1) = 1 +2* rand (1 ,1 , ” n o r m a l ” ) ; // Y = 2X + 1
24
15
where x i s g a u s s i a n d a t a
m_est = m_est + ((1/1000) * y ( i ) ) ; // e s t i m a t i o n by
averaging
var_est = var_est +((1/1000) *( y ( i ) - m_est ) ^2) ;
16
17 end
18 printf ( ’ E s t i m a t e d mean o f Y(=2X + 1 )
19
20
21
22
23
24
25
26
27
28
29
i s : Est mean=%f
’ , m_est )
printf ( ’ \n E s t i m a t e d v a r i a n c e o f Y) i s : E s t v a r i a n c e
=%f ’ , var_est )
// C a l c u l a t i o n o f t r u e mean o f Y
y_mean = integrate ( ’ ( 2 ∗ x +1) ∗ ( 1 / s q r t ( 2 ∗ %pi ∗ v a r i ) ∗ exp ( −(
x−m) ˆ 2 / ( 2 ∗ v a r i ) ) ) ’ , ’ x ’ , -100 ,100) ;
printf ( ’ \n True mean o f Y(=2X + 1 ) i s : True mean=%f ’
, y_mean )
// C a l c u l a t i o n o f t r u e v a r i a n c e o f Y
// f o r a f u n c t i o n l i k e Y = aX + b t h e v a r i a n c e o f Y
i s a ˆ2∗ V a r i a n c e o f X .
gs_mean = integrate ( ’ x ∗ ( 1 / s q r t ( 2 ∗ %pi ∗ v a r i ) ∗ exp ( −(x−m)
ˆ 2 / ( 2 ∗ v a r i ) ) ) ’ , ’ x ’ , -50 ,100) ;
gs_fsq = integrate ( ’ ( ( x ˆ 2 ) ∗ ( 1 / s q r t ( 2 ∗ %pi ∗ v a r i ) ∗ exp ( −(x
−m) ˆ 2 / ( 2 ∗ v a r i ) ) ) ) ’ , ’ x ’ , -50 ,100) ;
gs_var = gs_fsq - ( gs_mean ) .^2;
var_y = 2^2* gs_var ; // h e r e a = 2
printf ( ’ \n True v a r i a n c e o f Y(=2X + 1 ) i s :
T r u e v a r i a n c e=%f ’ , var_y )
30
31
// E x p e c t a t i o n o f Y i s E(Y)=E( 2X+1)=2E(X) +1. That ’ s
why a n s w e r i s 1 .
32 // True v a r i a n c e o f Y i n t h i s f o r m a t i s e q u a l t o a ˆ2∗
v a r i a n c e o f X.
33 // A s s i g n m e n t :
34 // 1 . Assume X i s u n i f o r m random v a r i a b l e b e t w e e n a
t o b . f i n d mean and v a r i a n c e .
25
Experiment: 7
Plot Joint density and
distribution function of sum of
two Gaussian random variable
(Z = X + Y).
Scilab code Solution 7.1 Joint Density and Distribution of Function of
two random varible
1
2
3
4
5
// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// P l o t J o i n t d e n s i t y and d i s t r i b u t i o n f u n c t i o n o f
sum o f two G a u s s i a n random v a r i a b l e ( Z = X + Y) .
6 clc ;
7 clear all ;
8 clf () ;
9
26
Figure 7.1: Joint Density and Distribution of Function of two random varible
Figure 7.2: Joint Density and Distribution of Function of two random varible
27
10 //PDF o f G a u s s i a n Random V a r i a b l e X
11 mean_x = 0; // mean o f f i r s t g a u s s i a n RV
12 sd_x = 1; // s t a n d a r d d e v i a t i o n o f f i r s t g a u s s s i a n RV
13 vari_x = sd_x .^2;
14
15 x = linspace ( -10 ,10 ,50) ; // g e n e r a t i n g l i n e a r l y s p a c e d
d a t a a s Random o u t p u t
16 X = ((1/( sqrt (2* %pi * vari_x ) ) ) * exp ( -0.5.*( x - mean_x )
.^2/ vari_x ) ) ; // f i n d i n g g a u s s i a n p d f o f a b o v e d a t a
17 subplot (2 ,1 ,1) ;
18 plot (x ,X , ’ r ’ )
19 xlabel ( ’ s a m p l e p o i n t s ’ ) ;
20 ylabel ( ’PDF o f X ’ ) ;
21 title ( ’PDF o f one Random V a r i a b l e ’ )
22
23
24 //PDF o f G a u s s i a n Random V a r i a b l e Y
25 mean_y = 0;
26 sd_y = 1;
27 vari_y = sd_y .^2;
28
29 y = linspace ( -10 ,10 ,50) ;
30 Y = ((1/( sqrt (2* %pi * vari_y ) ) ) * exp ( -0.5.*( y - mean_y )
31
32
33
34
35
36
37
38
.^2/ vari_y ) ) ;
subplot (2 ,1 ,2) ;
plot (y ,Y , ’ b ’ )
xlabel ( ’ s a m p l e p o i n t s ’ ) ;
ylabel ( ’PDF o f Y ’ ) ;
title ( ’PDF o f s e c o n d Random V a r i a b l e ’ )
// J o i n t p d f o f sum o f random v a r i a b l e X & Y
// When two IID random v a r i a b l e a r e summen up , t h e i r
J o i n t PDF i s c o n v o l u t i o n b e t w e e n i n d i v i d u a l p d f s
o f Random v a r i a b l e s
39 z = convol (X , Y ) ;
40 figure (2 , ” B a c k g r o u n d C o l o r ” ,[1 ,1 ,1]) ;
41 subplot (2 ,1 ,1) ; plot ( linspace ( -20 ,20 ,99) ,z ) // J o i n t
PDF
28
42 xlabel ( ’ s a m p l e p o i n t s ’ ) ;
43 ylabel ( ’PDF o f Z ’ ) ;
44 title ( ’ J o i n t d e n s i t y f u n c t i o n o f Z = X + Y ’ )
45 P = cdfnor ( ”PQ” , linspace ( -20 ,20 ,99) , mean ( z ) * ones ( z ) ,
46
47
48
49
50
51
52
53
sqrt ( variance ( z ) ) * ones ( z ) ) ;
set ( gca () ,” a u t o c l e a r ” ,” o f f ” )
subplot (2 ,1 ,2) ; plot2d (1: length ( P ) ,P ,3) ;
xlabel ( ’ s a m p l e p o i n t s ’ ) ;
ylabel ( ’ D i s t r i b u t i o n o f Z ’ ) ;
title ( ’ J o i n t d i s t r i b u t i o n f u n c t i o n o f Z = X + Y ’ )
set ( gca () ,” a u t o c l e a r ” ,” on ” )
// A s s i g n m e n t :
// Change mean and v a r i a n c e o f i n d i v i d u a l random
v a r i a b l e X and Y and o b s e r v e o u t p u t
29
Experiment: 8
Estimate the mean and
variance of a R.V. Z = X+Y.
Where X and Y are also
random variables.
Scilab code Solution 8.1 Estimation of mean and variance of sum of two
random variable
1
2
3
4
// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// Concept : E s t i m a t i o n o f mean and v a r i a n c e o f sum o f
two random v a r i a b l e Z = X + Y, where X and Y a r e
random v a r i a b l e .
5 // Above c o n c e p t i s e x p l a i n e d w i t h e x a m p l e a s
follows .
6 // Example : A l a r g e c i r c u l a r d a r t b o a r d i s s e t up w i t h
a ” b u l l s e y e ” a t t h e c e n t e r o f t h e c i r c l e , which
i s a t t h e c o o r d i n a t e ( 0 , 0 ) . A d a r t i s thrown a t
t h e c e n t e r but l a n d s a t (X, Y) a r e two d i f f e r e n t
G a u s s i a n random v a r i a b l e s . What i s a v e r a g e
d i s t a n c e o f t h e d a r t from t h e b u l l s e y e ? What i s
30
v a r i a n c e o f data ?
7 // D i s t a n c e from c e n t e r i s g i v e n a s s q r t (Xˆ2+Yˆ 2 )
8
9 clc ;
10 clear all ;
11 rand ( ’ s e e d ’ ,0) // s e t t i n g
s e e d o f random g e n e r a t o r t o
zero
12 m_est = 0;
13 for i = 1:1000
14
R (i ,1) = sqrt ( rand (1 ,1 , ’ n o r m a l ’ ) ^2+ rand (1 ,1 , ’
15
n o r m a l ’ ) ^2) ; // c a l c u l a t i o n o f d i s t a n c e from
center
m_est = m_est +(1/1000) * R ( i ) ; // e s t i m a t i o n o f mean
from d a t a
16 end
17 m_est
18 mprintf ( ’ Mean o f Sum o f Two Random v a r i a b l e
that i s
Mean o f Z = %f ’ , m_est )
19 v_est = variance ( R ) // v a r i a n c e c a l c u l a t i o n
20 mprintf ( ’ \n V a r i a n c e o f Sum o f Two Random v a r i a b l e
t h a t i s Mean o f Z = %f ’ , v_est )
31
Experiment: 9
Simulation of Central Limit
Theorem.
Scilab code Solution 9.1 Summation of two random variable leads to Gaussian density function
1
2
3
4
5
6
// O p e r a t i n g System : Windows XP o r l a t e r ,
// S c i l a b
: 5.3.3
// S i m u l a t i o n o f C e n t r a l L i m i t Theorem .
// ( Which s a y s t h a t i f we k e e p on a d d i n g i n d e p e n d a n t
Random V a r i a b l e s t h e n i t d e n s i t y f u n c t i o n
7 // a p p r o c h e s t o g a u s s i a n d i s t r i b u t i o n )
8 // h e r e two u n i f o r m RVs a r e added .
9
10
11
12
13
14
15
clc ;
clear all ;
clf () ;
n = 0:0.01:1;
x = zeros ( length ( n ) ,1) ;
i = 1:50; // l e n g t h o f Uniform r v 1
32
Figure 9.1: Summation of two random variable leads to Gaussian density
function
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
x ( i ) = 2;
subplot (3 ,3 ,1) ;
plot (n ,x , ’ r−d ’ )
xlabel ( ’ p d f ’ ) ; ylabel ( ’ p d f o f r v 1 ’ )
title ( ’ p d f o f r v 1 ’ ) ;
y = zeros ( length ( n ) ,1) ;
j = 1:50;
y ( j ) = 1*2; // l e n g t h o f Uniform r v 2
subplot (3 ,3 ,2) ;
plot (n ,y , ’ bo −. ’ )
xlabel ( ’ p d f ’ ) ;
ylabel ( ’ p d f o f r v 2 ’ )
title ( ’ p d f o f r v 2 ’ ) ;
z1 = convol (x , y ) ;
subplot (3 ,3 ,3)
//When two i n d e p e n d e n t RVs a r e added t h e i r j o i n t
33
density i s convolution of marginal density
plot ( z1 )
xlabel ( ’ r v v a l u e ’ ) ;
ylabel ( ’ p d f ’ )
title ( ’ j o i n t p d f o f r v 1 and r v 2 ’ ) ;
34
35
36
37
38
39 for i = 4:9 // a d d i n g r v 9 t i m e s
40
subplot (3 ,3 , i )
41
z1 = convol ( z1 , y ) ;
42
plot ( z1 )
43
xlabel ( ’ r v v a l u e ’ ) ;
44
ylabel ( ’ p d f ’ )
45 title ( ’ j o i n t p d f ’ ) ;
46 end
34