All physical subjects, involving random phenomena, something depending upon chance, naturally find their own way to theory of Statistics. Hence there arise relations between the results derived for hose random phenomena in different physical subjects and the concepts of Statistics. Convolution theorem has a variety of applications in field of Fourier transforms and many other situations, but it bears beautiful applications in field of statistics also .Here in this paper authors want to discuss some notions of Electrical Engineering in terms of convolution of some probability distributions.
Probability Distribution of Sum of Two Continuous Variables
1. Journal for Research| Volume 02| Issue 01 | March 2016
ISSN: 2395-7549
All rights reserved by www.journalforresearch.org 58
Probability Distribution of Sum of Two
Continuous Variables and Convolution
Rashmi R. Keshvani Yamini M. Parmar
Professor Assistant Professor
Department of Mathematics Department of Mathematics
Sarvajanik College of Engineering & Technology, Surat (Guj.)
India
Government Engineering College, Gandhi nagar (Guj),
India
Abstract
All physical subjects, involving random phenomena, something depending upon chance, naturally find their own way to theory
of Statistics. Hence there arise relations between the results derived for hose random phenomena in different physical subjects
and the concepts of Statistics. Convolution theorem has a variety of applications in field of Fourier transforms and many other
situations, but it bears beautiful applications in field of statistics also .Here in this paper authors want to discuss some notions of
Electrical Engineering in terms of convolution of some probability distributions.
Keywords: A Probability Distribution, An Uniform Probability Distribution, Central Limit Theorem, Convolution, Mean
And Variance of a Probability Distribution, Triangular Function, Unit Rectangle Function
_______________________________________________________________________________________________________
I. INTRODUCTION
Occurrence of resistance of a resistor, with its tolerance can be expressed as a probability distribution. In those circumstances,
what would be the resultant distribution describing occurrence of resistances of resistors having different resistance and different
tolerances, when combined in series? How means and variances are interrelated with that resultant distribution, is main focus of
this paper. In other words, to obtain probability distribution of sum of two random variables is main objective of this paper.
II. SOME BASIC CONCEPTS OF STATISTICS
A Probability Distribution, Its Mean and Variance:
A real valued function π(π₯) is said to be a probability distribution of a random variable π₯, if (1) π(π₯) β₯ 0, β π₯ β π and
β π(π₯) = 1πππ π₯ where π₯ is a discrete random variable. or (2) π(π₯) β₯ 0, β π₯ β π and β« π(π₯)ππ₯ = 1
β
ββ
where π₯ is a
continuous random variable. The mean of probability distribution π(π₯), denoted by ΞΌ, is defined as
(1) π = β π₯ π(π₯)πππ π₯ , if π₯ is discrete, or (2) π = β« π₯ π(π₯)ππ₯,
β
ββ
if π₯ is continuous. (1)
The variance of probability distribution π(π₯), denoted by π2
, is defined as
(1) π2
= β ( π₯ β π )2
π(π₯)πππ π₯ , if π₯ is discrete, or (2) π2
= β« ( π₯ β π )2β
ββ
π(π₯)ππ₯, if π₯ is continuous. [1] (2)
The Uniform Probability Distribution:
The uniform probability distribution, with parameters πΌ and π½, has probability distribution π(π₯) defined as
π(π₯) = {
1
π½βπΌ
ππ πΌ < π₯ < π½
0 πππ ππ€βπππ
(3)
III.THE CONCEPT OF CONVOLUTION
The convolution of two functions π(π₯) and π(π₯), [2], denoted by π(π₯) β π(π₯), or (π β π)(π₯) is defined as
π(π₯) β π(π₯) = β« π(π’)π(π₯ β π’)ππ’
β
ββ
(4)
One can easily check that the operation of convolution is commutative, associative and also distributive over addition.
Convolution has additive property also. That is
π β (π1 + π2)(π₯) = π(π₯) β (π1 + π2)(π₯) = π(π₯) β (π1(π₯) + π2(π₯)) = β« π(π’) (π1(π₯ β π’) + π2( π₯ β π’))ππ’
β
ββ
=
β« π(π’)π1(π₯ β π’)ππ’
ββ
β
+ β« π(π’)π2(π₯ β π’)ππ’
ββ
β
= ( π β π1)(π₯) + ( π β π2)(π₯)
That is π β (π1 + π2) = ( π β π1) + ( π β π2)
2. Probability Distribution of Sum of Two Continuous Variables and Convolution
(J4R/ Volume 02 / Issue 01 / 009)
All rights reserved by www.journalforresearch.org 59
IV.PROBABILITY DISTRIBUTION OF A SUM
Suppose two probability distributions π1(π₯1) and π2(π₯2) are given and it is required to determine probability distribution
π(π₯) of random variable π₯ = π₯1 + π₯2, sum of these two random variables. That is π(π₯) shows probability that sum of π₯1and π₯2
remains π₯. That is π(π₯) = π( π₯ = π₯1 + π₯2). As for example, suppose a large number of 100-ohm resisters , quoted as subject
to a 10 percent tolerance are drawn from an infinite supply containing equal numbers of resisters in any 1-ohm interval between
90 and 110 and none outside this range. Suppose that the stock of resisters has been drawn from a supply in which the frequency
of occurrence of resisters between π 1 and π 1 ππ 1 is π1(π 1)ππ 1. [2]
So π1(π 1) has to be probability distribution, more precisely, uniform probability distribution over interval 90 to 110, ensuring
that,
β« π1(π 1)
β
ββ
ππ 1 = 1, that is π1(π 1) = {
1
(110β90)
=
1
20
ππ 90 β€ π 1 β€ 110
0 πππ ππ€βπππ
Expressing π1(π 1) in terms of rectangle function, π1(π 1) =
1
20
β (
π 1β100
20
), where the rectangle function, denoted by β(π₯),
is defined as
β(π₯) = {
1 |π₯| < 1
2β
0 |π₯| > 1
2β
The mean of this distribution is π1 = β« π 1 π1(π 1)ππ 1 =
1
20
β« π 1 ππ 1 =
1
20
[
π 1
2
2
]
90
110
= 100
110
90
β
ββ
, and the variance π1
2
=
β« (π 1 β 100)2
π(π 1)ππ 1 =
1
20
β« (π 1 β 100)2
ππ 1 =
1
20
[
(π 1β100)3
3
]
90
110
=
100
3
110
90
β
ββ
Similarly for a stock of 50-ohm resisters, with 5-ohm tolerance, another probability distributionπ2(π 2), defined, as π2(π 2) =
{
1
(55β45)
=
1
10
ππ 45 β€ π 2 β€ 55
0 πππ ππ€βπππ
will describe the frequency of occurrence of resisters.
Expressing π2(π 2) also, in terms of rectangle function, π2(π 2) =
1
10
β (
π 2β50
10
) and the mean of π2(π 2), will be π2 = 50
and variance π2
2
=
25
3
If an electronic circuit, combining two of these stock resistors in series[2], and suppose the resulting distribution describing the
probability of occurrence of resistances, is denoted by π(π ), then here problem is to know, what mathematical relation, π(π )
bears with π1(π 1) and π2(π 2) when π = π 1 + π 2. It is clear that π = π 1 + π 2 will vary from 90 + 45 = 135 to 110 +
55 = 165, with mean equal to 150. It is also clear that the probability that π assumes a value less than or equal to135, that is
π(π β€ 135) = 0 and the probability that π assumes a value greater than or equal to 165 , that is π(π β₯ 165) = 0.
Probability that π assumes a particular value between 135 and 165, say, π = 140, that is π 1 + π 2 = 140, denoted by π(140)
will be that area where this sum assumes value 140.That is π 2 should remain 140 β π 1. Required area will be
π(140) =
1
20
β« π2(140 β π 1)ππ 1
110
90
=
1
20
β« π2(π 2)π
π 2=50
π 2=30
π 2 =
1
20
β«
1
10
π
π 2=50
π 2=45
π 2 =
1
40
,
As π 1 is nonzero if 90 β€ π 1 β€ 110 and π 2 is nonzero, if 45 β€ π 2 β€ 55.
One can understand that, this is nothing but convolution of π1(π 1) andπ2(π 2), that is
π(140) = (π1 β π2)(140) = β« π1(π 1)π2(140 β π 1)ππ 1
β
ββ
This is an instance of the basic convolution relation between the probability distribution describing the sum of two quantities
and the probability distributions of the given quantities. Now taking in general, let resistor π 1 have greater resistance π1 with
tolerance π1 , and resistor π 2 have smaller resistance π2 with tolerance π2. As π1 > π2 and, as π1 > π1, π2 > π2, there is
no loss of generality any, if we assume π1 > π2 > π1 > π2 > 0.
In terms of rectangle functions π 1 =
1
2π1
β (
π 1βπ1
2π1
) and π 2 =
1
2π2
β (
π 2βπ2
2π2
)
It is quite obvious that π1 = β« π 1 π1(π 1)ππ 1
β
ββ
=
1
2π1
β« π 1 ππ 1
π1+π1
π1βπ1
=
4π1 π1
4π1
= π1
π2 = β« π 2 π2(π 2)ππ 2
β
ββ
=
1
2π2
β« π 2 ππ 2
π2+π2
π2βπ2
=
4π2 π2
4π2
= π2
π1
2
= β« (π 1 β π1)2
π1(π )ππ 1
β
ββ
=
1
2π1
β« (π 1 β π1)2
ππ 1
π1+π1
π1βπ1
=
1
2π1
[
(π 1 β π1 )3
3
] π1βπ1
π1+π1
=
π1
2
3
π2
2
= β« ( π 2 β π2)2
π2(π )ππ
β
ββ
=
1
2π2
β« ( π 2 β π2)2
ππ
π2+π2
π2βπ2
=
1
2π2
[
(π 2 β π2 )3
3
] π2βπ2
π2+π2
=
π2
2
3
To determine π(π ), following cases are to be considered separately,
If π β€ π1 + π2 β π1 β π2 or π β₯ π1 + π2 + π1 + π2, then π(π ) = 0
If (π1 + π2) β (π1 + π2) < π < (π1 + π2) β π1 + π2
π(π ) = β« π1(π 1)π2(π β π 1)ππ 1 =
1
2π1
β« π2(π β π 1)ππ 1 =
1
2π1
β« π2(π 2)ππ 2
π β(π1βπ1)
π β(π1 + π1 )
π1+ π1
π1βπ1
β
ββ
=
1
4π1 π2
β« ππ 2
π β(π1βπ1)
π2βπ2
5. Probability Distribution of Sum of Two Continuous Variables and Convolution
(J4R/ Volume 02 / Issue 01 / 009)
All rights reserved by www.journalforresearch.org 62
Ξ(π₯) = {
0 ππ |π₯| > 1
1 β |π₯| ππ |π₯| β€ 1
In this case, π1 = π2, π1 = π2 and
β« π(π )ππ = β«
1
4π2 (π β 2( π β π))ππ + β«
1
4π2
(2( π + π) β π )ππ
2(π+π)
2π
2π
2(πβπ)
β
ββ
=
1
4π2 {[
(π β2(πβπ))
2
2
]
2(πβπ)
2π
+
[β
(2(π+π)βπ )2
2
]
2π
2(π+π)
} =
1
4π2
{ 2π2
+ 2π2} = 1.
π = β« π π(π )ππ
β
ββ
= { β«
1
4π2 π (π β 2( π β π))ππ
2π
2(πβπ)
+ β«
1
4π2 π (2(π + π) β π )ππ
2(π+π)
2π
} =
1
4π2 { [
π 3
3
]
2(πβπ)
2π
β 2(π β π) [
π 2
2
]
2(πβπ)
2π
+
2(π + π) [
π 2
2
]
2π
2(π+π)
β [
π 3
3
]
2π
2(π+π)
} =
1
4π2 {
16π3
3
β
8(πβπ)3
3
+ 4(π β π)3
+ 4(π + π)3
β
8(π+π)3
3
β 8π3
}
=
1
4π2
{
β8π3
3
+
4(π β π)3
3
+
4(π + π)3
3
} =
1
3π2
{β2π3 + 2π3 + 6ππ2} = 2π = 2π1
π2
= β« ( π β 2π)2
π(π )ππ
β
ββ
= β« ( π β 2π)2
1
4π2
(π β 2( π β π))ππ + β«
1
4π2
( π β 2π)2(2(π + π) β π )ππ
2(π+π)
2π
2π
2(πβπ)
=
1
4π2
{ β« (( π β 2π)3 + 2π( π β 2π)2)ππ + β« ((2π β π )3 + 2π(2π β π )2)ππ
2(π+π)
2π
2π
2(πβπ)
}
=
1
4π2
{ [
(π β 2π)4
4
]
2(πβπ)
2π
+ 2π [
(π β 2π)3
3
]
2(πβπ)
2π
β [
(2π β π )4
4
]
2π
2(π+π)
β 2π [
(2π β π )3
3
]
2π
2(π+π)
}
=
1
4π2
{4π4
+ 16
π4
3
β 4π4
+ 16
π4
3
} =
8π2
3
= 2 π1
2
By (5) and (8), it can be observed that the original distributions had been more rounded by convolution. If more elements were
connected in series then the tendency toward smoothness would be more advanced. If a large number of functions are convolved
together, the resultant may be very smooth and as the number increases indefinitely, the resultant may approach Gaussian
form.[2] In Statistics Gaussian distribution is referred to as βnormal distribution with zero mean and standard deviation oneβ,
where the normal distribution is defined as
π( π β€ π₯) =
1
πβ2π
β« πβ
1
2
(
π₯βπ
π
)2
π₯
ββ
ππ₯ ,
Where π is a continuous random variable. The rigorous statement of the above stated tendency of protracted convolution is
nothing but the theorem known as the central-limit theorem.[1].
VII. CENTRAL LIMIT THEOREM
If πΜ is the mean of a sample of size π taken from a population having the mean π and the finite variance π2
, then π§ =
πΜ βπ
π
βπ
β
is
a random variable whose distribution function approaches that of the standard normal distribution as π β β. [1]
From the central limit theorem, it follows that if several, n functions are convolved together or a function is self-convolved n
times, the result approaches Gaussian distribution.[3] Therefore it follows that if several random quantities are added, then the
frequency distribution of the sum will approach a Gaussian distribution.
VIII. CONCLUSION
ο Distribution of sum of two random variables is convolution of distributions of those random variables. Composite
resistance of two resistors, combined in series, is obtained by convolution of distributions of those resistors.
ο Convolution has additive property of means of the components. The mean value of the composite resistance of two
resistors, combined in series, is the sum of the means of those two resistors.
ο Variances are also additive under convolution. Variance of composite resistance of two resistors, combined in series,
is addition of variances of those two resistors.
REFERENCES
[1] Richard A. Johnson, Miller & Freundβs βProbability and Statistics for Engineersβ, Sixth edition
[2] Ronald N. Bracewell, βThe Fourier transform and its applicationsβ, International edition 2000, McGrow-Hill Education.
[3] Rashmi R. Keshvani, Yamini M Parmar, βAn approach to various Probability Distributions through Convolution.β, international journal of Physical,
Chemical & Mathematical Sciences. Vol-3, pp-1-8, July-Dec- 2014