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In the Name of the Most High. Performance Evaluation of Computer Systems. Introduction to Probabilities: Discrete Random Variables. By Behzad Akbari Tarbiat Modares University Spring 2009. These slides are based on the slides of Prof. K.S. Trivedi (Duke University). Random Variables. - PowerPoint PPT Presentation
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1
Performance Evaluation of Computer Systems
By
Behzad Akbari
Tarbiat Modares University
Spring 2009
Introduction to Probabilities: Discrete Random Variables
These slides are based on the slides of Prof. K.S. Trivedi (Duke University)
In the Name of the Most High
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Random Variables
Sample space is often too large to deal with directly Recall that in the sequence of Bernoulli trials, if we
don’t need the detailed information about the actual pattern of 0’s and 1’s but only the number of 0’s and 1’s, we are able to reduce the sample space from size 2n
to size (n+1). Such abstractions lead to the notion of a random
variable
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Discrete Random Variables
A random variable (rv) X is a mapping (function) from the sample space S to the set of real numbers
If image(X ) finite or countable infinite, X is a discrete rv Inverse image of a real number x is the set of all sample points that
are mapped by X into x:
It is easy to see that
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Probability Mass Function (pmf)
Ax : set of all sample points such that,
pmf
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pmf Properties
Since a discrete rv X takes a finite or a countable infinite set
values,
the last property above can be restated as,
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Distribution Function
pmf: defined for a specific rv value, i.e., Probability of a set
Cumulative Distribution Function (CDF)
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Distribution Function properties
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Equivalence: Probability mass function Discrete density function(consider integer valued random variable)
cdf:
pmf:
)( kXPpk
x
kkpxF
0)(
Discrete Random Variables
)1()( kFkFpk
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Common discrete random variables
Constant Uniform Bernoulli Binomial Geometric Poisson
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Constant Random Variable
pmf
CDF
c
1.0
1.0
c
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Discrete Uniform Distribution
Discrete rv X that assumes n discrete value with equal probability 1/n
Discrete uniform pmf
Discrete uniform distribution function:
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Bernoulli Random Variable
RV generated by a single Bernoulli trial that has a binary valued outcome {0,1} Such a binary valued Random variable X is called the indicator or Bernoulli random variable so that
Probability mass function:
)0(1
)1(
XPpq
XPp
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Bernoulli Distribution
CDF
x0.0 1.0
q
p+q=1
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Binomial Random Variable
Binomial rv a fixed no. n of Bernoulli trials (BTs) RV Yn: no. of successes in n BTs Binomial pmf b(k;n,p)
Binomial CDF
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Binomial Random Variable
In fact, the number of successes in n Bernoulli trials can be seen as
the sum of the number of successes in each trial:
where Xi ’s are independent identically distributed Bernoulli random
variables. nn XXXY ...21
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Binomial Random Variable: pmf
pk
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0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10
x
CD
FBinomial Random Variable: CDF
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Reliability of a k out of n system
Series system:
Parallel system:
Applications of the binomial
njnjn
j
njparallel
njnjn
nj
njseries
jnjn
kj
nj
n
kjkofn
RRRRnbR
RRRRnnbR
RRRnjbRnkBR
]1[1]1[])[(),;0(1
][]1[])[(),;(
]1[])[(),;(),;1(1
1
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Transmitting an LLC frame using MAC blocks p is the prob. of correctly transmitting one block Let pK(k) be the pmf of the rv K that is the number of
LLC transmissions required to transmit n MAC blocks correctly; then
Applications of the binomial
nknkK
nnK
nK
ppkp
and
ppp
ppnnbp
])1(1[])1(1[)(
])1(1[)2(
),;()1(
1
2
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Geometric Distribution
Number of trials upto and including the 1st success.
In general, S may have countably infinite size
Z has image {1,2,3,….}. Because of independence,
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Geometric Distribution (contd.)
Geometric distribution is the only discrete distribution that exhibits MEMORYLESS property.
Future outcomes are independent of the past events. n trials completed with all failures. Y additional trials are performed
before success, i.e. Z = n+Y or Y=Z-n
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Geometric Distribution (contd.)
Z rv: total no. of trials upto and including the 1st success. Modified geometric pmf: does not include the successful
trial, i.e. Z=X+1. Then X is a modified geometric random variable.
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The number of times the following statement is executed:
repeat S until B
is geometrically distributed assuming …. The number of times the following statement is
executed:
while B do S
is modified geometrically distributed assuming ….
Applications of the geometric
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Negative Binomial Distribution
RV Tr: no. of trials until rth success.
Image of Tr = {r, r+1, r+2, …}. Define events: A: Tr = n B: Exactly r-1 successes in n-1 trials. C: The nth trial is a success.
Clearly, since B and C are mutually independent,
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Poisson Random Variable
RV such as “no. of arrivals in an interval (0,t)” In a small interval, Δt, prob. of new arrival= λΔt. pmf b(k;n, λt/n); CDF B(k;n, λt/n)=
What happens when
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Poisson Random Variable (contd.)
Poisson Random Variable often occurs in situations, such as, “no. of packets (or calls) arriving in t sec.” or “no. of components failing in t hours” etc.
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Poisson Failure Model
Let N(t) be the number of (failure) events that occur in the time interval (0,t). Then a (homogeneous) Poisson model for N(t) assumes:
1.The probability mass function (pmf) of N(t) is:
Where > 0 is the expected number of event occurrences per unit time
2.The number of events in two non-overlapping intervals are mutually independent
/ !k
tP N t k kt e
,2,1,0k
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Note:
For a fixed t, N(t) is a random variable (in this case a discrete random variable known as the Poisson random variable). The family {N(t), t 0} is a stochastic process, in this case, the homogeneous Poisson process.
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Poisson Failure Model (cont.)
The successive interevent times X1, X2, … in a homogenous Poisson model, are mutually independent, and have a common exponential distribution given by:
To show this:
Thus, the discrete random variable, N(t), with the Poisson distribution, is related to the continuous random variable X1, which has an exponential distributionThe mean interevent time is 1/, which in this case is the mean time to failure
eXttP 11 0t
tetNPtXP )0)(()( 1
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Probability mass function (pmf) (or discrete density function):
Distribution function (CDF):
k!
)( )(
ktektNPp t
k
k!)(
0
k
x
k
t texF
Poisson Distribution
31
pk
t=1.0
Poisson pmf
32
t1 2 3 4 5 6 7 8 9 10
0.5
0.1
CDF1
t=1.0
Poisson CDF
33
t=4.0
pk
t=4.0
Poisson pmf
34
t
CDF
1 2 3 4 5 6 7 8 9 10
0.5
0.1
1
t=4.0
Poisson CDF
35
Probability Generating Function (PGF)
Helps in dealing with operations (e.g. sum) on rv’s Letting, P(X=k)=pk , PGF of X is defined by,
One-to-one mapping: pmf (or CDF) PGF See page 98 for PGF of some common pmfs
36
Discrete Random Vectors
Examples: Z=X+Y, (X and Y are random execution times) Z = min(X, Y) or Z = max(X1, X2,…,Xk)
X:(X1, X2,…,Xk) is a k-dimensional rv defined on S
For each sample point s in S,
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Discrete Random Vectors (properties)
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Independent Discrete RVs
X and Y are independent iff the joint pmf satisfies:
Mutual independence also implies:
Pair wise independence vs. set-wide independence
39
Discrete Convolution
Let Z=X+Y . Then, if X and Y are independent,
In general, then,