Download Bernoulli Random Variable

Bernoulli Random Variable
Random variable X is the indicator or
Bernoulli random variable so that:
!
!
Probability Mass Function:
!
Mean E[X]:
p ! P( X ! 1)
q ! 1 " p ! P( X ! 0)
E [ X ] ! 0.q # 1. p ! p ! P( X ! 1)
E[ X 2 ] ! 0.q # 1. p ! p
Var[ X ] ! p " p 2 ! pq
Copyright © 2006 by K.S. Trivedi
40
Binomial Distribution
!
Let Y be a binomial rv with parameter n, p.
pk ! P (Y ! k ) ! C(n,k) p k ( 1-p)n- k
!
So Y can be written as the sum of n mutually
independent Bernoulli rv’s X1,X2 ..Xn.
!
So using the linearity property of the expectation
we have
n
E[Y ] ! $ E[ X i ] ! np
i !1
!
n
Similarly variance is given by Var[Y ] ! $Var[ X i ] ! npq
i !1
Copyright © 2006 by K.S. Trivedi
41
Geometric Distribution
!
The pmf is given by
!
The mean is computed as
Copyright © 2006 by K.S. Trivedi
42
Poisson Distribution
probability mass function (pmf):
!
Mean
!
i
(
(
)
p X (i ) ! e "(
, 0 ' i ' &, ( % 0
i!
&
i( i "(
E[X] : E[ X ] ! $ e
i ! 0 i!
E[X ] ! (
The variance is also given by
Var [ X ] ! (
!
Copyright © 2006 by K.S. Trivedi
43
Exponential Distribution (Very
important)
!
Distribution Function:F )t * ! 1 " e
", t
t+0
Density Function:
f )t * ! , e " , t
!
Reliability:
!
Failure Rate (CFR):
R ) t * ! e " ,t
t+0
f )t *
h)t * !
!,
R )t *
!
!
t+0
Failure rate is age-independent (constant)
1
!
Mean Time to Failure: MTTF ! ,
Copyright © 2006 by K.S. Trivedi
44
Exponential Distribution (Very
important)
!
!
The rate parameter is not exponential; it is
constant or age independent
Mean (expected) value is the reciprocal of
its rate parameter
Copyright © 2006 by K.S. Trivedi
45
Weibull Distribution
!
Failure Rate: h)t * !
!
IFR for ( % 1 DFR for ( - 1
!
Expectation (MTTF):
f (t )
R (t )
! ,( t ( "1
t+0
1
3 1 0( 3 1 0
E[X ] ! 1 . 411 # .
2,/ 2 ( /
!
Shape parameter ( and scale parameter,
Copyright © 2006 by K.S. Trivedi
46
Continuous Uniform Distribution
!
The density function is given by
!
Expectation is calculated as
!
The kth moment is computed as:
!
Therefore variance becomes
Copyright © 2006 by K.S. Trivedi
47
Hypoexponential Distribution
!
!
if X1,X2, . . . , Xn are mutually
independent exponentially distributed
random variables with parameters
then
is
hypoexponentially distributed with
parameters
So using linearity property of
expectation we have
Copyright © 2006 by K.S. Trivedi
48
Hypoexponential Distribution
(contd.)
•Expectation as
•And Variance as
•The coefficient of variation:
Copyright © 2006 by K.S. Trivedi
49
Hyperexponential Distribution
!
The density in this case is given by
!
Then the mean becomes
Copyright © 2006 by K.S. Trivedi
50