CS723 - Probability
and
Stochastic Processes
CS723 - Probability
and
Stochastic Processes
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Lecture 30: Sums of Independent Random Variables and Characteristic Functions, Slides of Probability and Stochastic Processes
The concept of sums of independent random variables and their characteristic functions in the context of probability and stochastic processes. The lecture covers the pdfs of sums of iid random variables, the convolution integral, and the relationship between characteristic functions and fourier transforms. Students will learn how to find the pdf of the sum of n iid random variables using n-fold convolution.
Typology: Slides
2011/2012
1 / 15
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CS723 - Probability
and
Stochastic Processes
CS723 - Probability
and
Stochastic Processes
- Lecture No. 30Lecture No.
PDF of Z=X1+X2 = X+X
2X
- Instead of writing Z = X+X, we could have
used Z = X + Y with f
X
(x) = f
Y
(y)
- We are adding two values from two
random variables X
1
and X
2
which have
same PDF as the PDF of X
- Using the PDF via CDF method, we can
find the PDF of Z=X+X
- The joint PDF of X
1
and X
2
is defined over
whole x
1
-x
2
plane
PDF of Z = X+X
2X
PDF of Z = X+X+X+…
) x ( f ) x ( f
(z)
f
X
X
Z
2
) x ( f ) x ( f ) x ( f
(z)
f
X
X
X
Z
3
Find PDF of Z
2
= X + X by convolving PDF of
X with itself
Find PDF of Z
3
= X + X + X = X + Y by
convolving PDF of Y with PDF of X
)
x (
f
...
) x ( f ) x ( f
(z)
f
PDF of sum of N iid random variables is
obtained from N-fold convolution
PDF’s of Sum of iid RV’s
PDF’s of X , X+X , X+X+X , and X+X+X+X+X
Characteristic Function
)
x
sin(
jE
)
x
cos(
E
)
x
sin( j
)
x
cos(
E
e
E
)
(
x
j
X
Characteristic function of a random variable X
is expected value of g(x) = e
j ω
x
X
) = E[e
j ω
x
] gives an expression that is a
function of parameter
Using Euler’s identity, we can writeCharacteristic function of a random variable X
depends upon the PDF of X
Characteristic function of a random variable is
Characteristic Function
a
sin(
dx
e
a
a a
x j
X
For a uniformly distributed random variable
over [-a,a] the characteristic functions is
For an exponential random variable
j
dx
e
e
x
0
x j
X
For a Gaussian random variable N(0,1)
2
x^2
x
j
X
2
2
e
dx
e
1
e
)
(
Characteristic Function
of Sum of i.i.d. RV’s
If X, X
1
, X
2
, X
3
are i.i.d. and Z = X
1
+X
2
+X
3
, then
N
3 X X X X Z
X
X
X
Z
) ( ) ( ) ( ) ( ) (
) x ( f ) x ( f ) x ( f ) z ( f
If X, X
1
, X
2
, X
3
, … , X
N
are i.i.d. random
variables and Z = X
1
+ X
2
+ X
3
+ … + X
N
Sum of Gaussian RV’s
2
2
2
e
)
e
)(
e ( ) ( ) ( ) (
2
2
X
X
Z
Sum of two iid Gaussian N(0,1) random
variables has a characteristic function
Sum of N iid Gaussian N(0,1) random
variables has a characteristic function
2 N N 2 N X Z
2
2
e ) e ( ) ( ) (