1 continuous random variables continuous random variable let x be such a random variable takes on...
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Continuous random variables Continuous random variable
Let X be such a random variable Takes on values in the real space
(-infinity; +infinity)
(lower bound; upper bound)
Instead of using P(X=i) Use the probability density function
fX (t)
Or fX (t)dt
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Cumulative function of continuous r.v.
The relationship between the Cumulative distribution of continuous r.v. and fX
=>
Properties for CDF
t
XX dssftF )()(
)()( tftFdt
dXX
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Distribution function: properties
Properties for pdf
0][;0)(][)
][1)(][)
1)()
0)()
tXPdtdttfdttXtPd
tXPdttftXPc
dttfb
tfa
X
t
X
X
X
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Uniform random variable
X is a uniform random variable
Mean:
Variance:
];[;0
;1
)(
];[
bat
btaabtf
baX
X
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Exponential distribution
Exponential distribution is the foundation of most of the stochastic processes
Makes the Markov processes ticks
is used to describe the duration of sthg CPU service
Telephone call duration
Or anything you want to model as a service time
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Exponential random variable
A continuous r.v. X Whose density function is given for
is said to be an exponential r.v. with parameter λ
Mean: and variance:
0;0
0;)(
][1;0
t
tetf
XE
t
X
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Link between Poisson and Exponential
If the arrival process is Poisson # arrivals per time unit follows the Poisson distribution
With parameter λ
=> inter-arrival time is exponentially distributed With mean = 1/ λ = average inter-arrival time
Time
0 T
Exponentiallydistributed with 1/ λ
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Proof
Number of arrivals in a t-second interval Follows the Poisson distribution with parameter
Let denote random time of first arrival
=> T is exponentially distributed
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Example
Suppose that amount of time you spend in bank is exponential with mean 10 min
What is the probability you spend more than 5 min in bank?
What is the probability you spend more than 15 min Given that you are still in bank after 10 min?
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Hyper-exponential distributions H2 Hn
Advantage Allows a more sophisticated representation
Of a service time
While preserving the exponential distribution And have a good chance of analyzing the problem
λ1
λ2
p
1-p
.
.
λ1
λ2
λn
p1
p2
pn
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Further properties of the exponential distribution
If are independent exponential r.v. With mean , then the pdf of is:
Gamma distribution with parameters n and
If and are independent exponential r.v. With mean and =>
𝑓 𝑋 1+𝑋 2+…+𝑋 𝑛(𝑡 )=𝜆𝑒−𝜆𝑡
(𝜆𝑡 )𝑛−1
(𝑛−1 )!
𝑃 (𝑋 1< 𝑋 2)=𝜆1
𝜆1+𝜆2
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Further properties of the exponential distribution (ct’d)
X1 , X2 , …, Xn independent r.v. Xi follows an exponential distribution with
Parameter λi => fXi (t) = λi eλit
Define X = min{X1, X2, …, Xn} is also exponentially distributed
Proof fX(t) = ?
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Joint distribution functions Discrete case
One variable (pmf) P(X=i)
Joint distribution P(X1=i1, X2=i2, …, Xn=in)
Continuous case One variable (pdf)
fX(t)
Joint distribution fX1, X2,…Xn, (t1,t2,.., tn)
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Independent random variables The random variables X1 , X2
Are said to be independent if, for all a, b
Example Green die: X1
Red die: X2
X3 = X1 + X2
X3 and X1 are dependent or independent?
)().(),( bYPaXPbYaXP
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Marginal distribution
Joint distribution Discrete case
P(X1=i, X2=j), for all i, j in S1xS2
=>
Continuous case fX1,X2(t1,t2), for all t1,t2
=>
2
),()( 211Sj
jXiXPiXP
221,1 ),()(211
dtttftf XXX
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Expectation of a r.v.: the continuous case
X is a continuous r.v. Having a probability density function f(x)
The expected value of X is defined by
Define g(X) a function of r.v. X
dxxxfXE X )(][
dxxfxgXgE X )().()]([
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Expectation of a r.v.: the continuous case (cont’d)
X1, X2, …, Xn: dependent or independent
Example:
)(...)()()...( 2121 nn XEXEXEXXXE
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Variance, auto-correlation, & covariance Variance
Continuous case
If are independent r.v. =>
If X and Y are correlated r.v.:
Autocorrelation:
Covariance
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Conditional probability and conditional expectation: d.r.v.
X and Y are discrete r.v. Conditional probability mass function
Of X given that Y=y
Conditional expectation of X given that Y=y
)(
),(
)(
),(
)|()|(|
yp
yxp
yYP
yYxXP
yYxXPyxp
Y
YX
x
yYxXPxyYXE )|(.]|[
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Conditional probability and expectation: continuous r.v.
If X and Y have a joint pdf fX,Y(x,y) Then, the conditional probability density function
Of X given that Y=y
The conditional expectation Of X given that Y=y
)(
),()|(| yf
yxfyxf
YYX
dxyxfxyYXE YX )|(.]|[ |
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Computing expectations by conditioning Denote
E[X|Y]: function of the r.v. Y Whose value at Y=y is E[X|Y=y]
E[X|Y]: is itself a random variable Property of conditional expectation
if Y is a discrete r.v.
if Y is continuous with density fY (y) =>
]]|[[][ YXEEXE
y
yYPyYXEYXEEXE ][].|[]]|[[][
dyyfyYXEXE Y )(]|[][
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