2. Review of Probability and Statistics

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2. Review of Probability and Statistics
Refs Law Kelton, Chapter 4
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Random Variables

• Experiment a process whose outcome is not known
  with certainty
• Sample space S all possible outcomes of an
  experiment
• Sample point an outcome (a member of sample
  space S)
• Example in coin flipping, SHead,Tail, Head
  and Tail are outcomes

• Random variable a function that assigns a real
  number to each point in sample space S
• Example in flipping two coins
• If X is the random variable number of heads
  that occur
• then X1 for outcomes (H,T) and (T,H), and X2
  for (H,H)

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Random Variables Notation

• Denote random variables by upper case letters X,
  Y, Z, ...
• Denote values of random variables by lower case
  letters x, y, z,
• The distribution function (cumulative
  distribution function) F(x) of random variable X
  for real number x is
• F(x) P(X?x) for -?ltxlt?
• where P(X?x) is the probability associated with
  event X?x
• F(x) has the following properties
• 0?F(x) ?1 for all x
• F(x) is nondecreasing i.e. if x1?x2 then F(x1)
  ? F(x2)

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Discrete Random Variables

• A random variable is discrete if it takes on at
  most a countable number of values
• The probability that random variable X takes on
  value xi is given by
• p(xi) P(Xxi) for I1,2,
• and
• p(x) is the probability mass function of discrete
  random variable X
• F(x) is the probability distribution function of
  discrete random variable X

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Continuous Random Variables
X is a continuous random variable if there exists
a non-negative function f(x) such that for any
set of real numbers B, f(x) is the probability
density function for the continuous RV X F(x)
is the probability distribution function for the
continuous RV X
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Distribution and density functions of a
Continuous Random Variable
Given an interval I a,b
f(x)
a
b
x
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Joint Random Variables
In the M/M/1 queuing system, the input can be
represented as two sets of random
variables arrival times of customers A1, A2, ,
An and service times of customers S1, S2, ,
Sn The output can be a set of random
variables delays in queue of customers D1, D2,
, Dn The Ds are not independent.
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Jointly discrete Random Variables
Consider the case of two discrete random
variables X and Y, the joint probability mass
function is p(x,y) P(Xx,Yy) for all x,y X
and Y are independent if p(x,y) pX(x)pY(y) for
all x,y these are the marginal probability mass
functions
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Jointly continuous Random Variables
Consider the case of two continuous random
variables X and Y. X and Y are jointly continuous
random variables if there exists a non-negative
function f(x,y) (the joint probability density
function of X and Y) such that for all sets of
real numbers A and B, X and Y are independent
if fX(x) and fY(y) are called the marginal
probability density functions
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Measuring Random Variables mean and median
Consider n random variables X1, X2, , Xn The
mean of the random variable Xi (i1,2,,n) is
The mean is the expected value, and is a measure
of central tendency. The median, x0.5 , is the
smallest value of x such that FX(x) ? 0.5 For a
continuous random variable, F(x0.5) 0.5
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Measuring Random Variables variance
The variance of the random variable Xi , denoted
by ?i2 or Var(Xi) is ?i2 E(Xi-?i)2 E(Xi 2)
- ?i2 (this is a measure of dispersion)
small ?2
large ?2
?
?
Some properties of variance are Var(X) ?
0 Var(cX) c2 Var(X) If the Xis are
independent, then Var(X Y) Var(X) Var(Y) or
Standard deviation is
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Variance is not dimensionless
e.g. if we collect data on service times at a
bank machine with mean 0.5 minutes, variance 0.1
minute, the same data in seconds will give mean
30 seconds, variance 360 seconds i.e., variance
is dependent on scale!
602
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Measures of dependence
The covariance between random variables Xi and Xj
where i1,,n and j1,,n is Cij Cov(Xi , Xj)
E(Xi - ?i)(Xj- ?j) EXi Xj - ?i ?j Some
properties of covariance Cij Cji if ij, then
Cij Cji Cii Cjj ?i2 if Cij gt 0, then Xi
and Xj are positively correlated Xi gt ?i and
Xj gt ?j tend to occur together, and Xi lt ?i and
Xj lt ?j tend to occur together if Cij lt 0, then
Xi and Xj are negatively correlated Xi gt ?i
and Xj lt ?j tend to occur together, and Xi lt ?i
and Xj gt ?j tend to occur together Cij is NOT
dimensionless, i.e. the value is influenced by
the scale of the data.
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A dimensionless measure of dependence
Correlation is a dimensionless measure of
dependence
Var(a1X1 a2X2 ) a12Var(X1)2a1a2Covar(X1, X2)
a22Var(X1) Var(X-Y) Var(X) Var(Y)
2Cov(X,Y) Var(XY) Var(X) Var(Y) 2Cov(X,Y)
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• The covariance between the random variables X and
  Y, denoted by Cov(X, Y), is defined by
• Cov(X, Y) EX - E(X)Y - E(Y)
• E(XY) - E(X)E(Y)
• The covariance is a measure of the dependence
  between X and Y. Note that Cov(X, X) Var(X).

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• Definitions
• Cov(X, Y) X and Y are
• 0 uncorrelated
• gt 0 positively correlated
• lt 0 negatively correlated
• Independent random variables are also
  uncorrelated.

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• Note that, in general, we have
• Var(X - Y) Var(X) Var(Y) -
• 2Cov(X, Y)
• If X and Y are independent, then
• Var(X - Y) Var(X) Var(Y)
• The correlation between the random variables X
  and Y, denoted by Cor(X, Y), is defined by

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• It can be shown that
• -1 ? Cor(X, Y) ? 1

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4.2. Simulation Output Data and Stochastic
Processes

• A stochastic process is a collection of "similar"
  random variables ordered over time all defined
  relative to the same experiment. If the
  collection is X1, X2, ... , then we have a
  discrete-time stochastic process. If the
  collection is X(t), t ? 0, then we have a
  continuous-time stochastic process.

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• Example 4.3 Consider the single-server queueing
  system of Chapter 1 with independent interarrival
  times A1, A2, ... and independent processing
  times P1, P2, ... . Relative to the experiment
  of generating the Ai's and Pi's, one can define
  the discrete-time stochastic process of delays in
  queue D1, D2, ... as follows
• D 1 0
• Di 1 maxDi Pi - Ai 1, 0 for i 1, 2, ...

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• Thus, the simulation maps the input random
  variables into the output process of interest.
• Other examples of stochastic processes
• N1, N2, ... , where Ni number of
• parts produced in the ith hour
• for a manufacturing system
• T1, T2, ... , where Ti time in
• system of the ith part for a
• manufacturing system

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• Q(t), t ? 0, where Q(t) number of
  customers in queue at time t
• C1, C2, ... , where Ci total cost in
  the ith month for an inventory system
• E1, E2, ... , where Ei end-to-end delay
  of ith message to reach its destination in a
  communications network
• R(t), t ? 0, where R(t) number of red
  tanks in a battle at time t

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• Example 4.4 Consider the delay-in-queue process
  D1, D2, ... for the M/M/1 queue with utilization
  factor ?. Then the correlation function ?j
  between Di and Dij is given in Figure 4.8.

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?j
? 0.9
1.0
0.9
0.8
0.7
0.6
0.5
? 0.5
0.4
0.3
0.2
0.1
j
0
1
2
3
4
5
6
9
10
7
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Figure 4.8. Correlation function ?j of the
process D1, D2, ... for the M/M/1 queue.
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Estimating distribution parameters
Assume that X1, X2, , Xn are independent,
identically distributed (IID) random variables
with finite population mean ?, and finite
population variance ?2. We would like to estimate
? and ?2 The sample mean is This is an
unbiased estimator of ?, i.e. also, for a very
large number of independent calculations of
, the average of the will
tend to ?
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Estimating variance
The sample variance is This is an
unbiased estimator of ?, i.e. Es2(n)
?2 this is a measure of how well
estimates ?
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Two interesting theorems
Central Limit Theorem For large enough n,
tends to be normally distributed with mean ?
and variance ?2/n This means that we can assume
that the average of a large numbers of random
samples is normally distributed - very
convenient. Strong Law of Large Numbers For an
infinite number of experiments, each resulting in
, for sufficiently large n, will
be arbitrarily close to ? for almost all
experiments. This means that we must choose n
carefully every sample must be large enough to
give a good .
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Area 0.68
Figure 4.7. Density function for a N(?, ?2)
distribution.
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Confidence Intervals
f(x)
1-?
x
0
Interpretation of confidence intervals If one
constructs a very large number of independent
100(1-?) confidence intervals, each based on n
observations, with n sufficiently large, then the
proportion of these confidence intervals that
contain ? should be 1-?. What is n sufficiently
large? The more non-normal the distribution of
Xi , the larger n must be to get good coverage
(1- ?).
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An exact confidence interval
If the Xis are normal random variables, then tn
has a t distribution with n-1 degrees of freedom
(df), and an exact 100(1-?) confidence interval
for ? is This is better than zs confidence
interval, because it provides better coverage (1-
?) for small n, and converges to the z confidence
interval for large n. Note this estimator
assumes that the Xis are normally distributed.
This is reasonable if the central limit theorem
applies i.e. each Xi is the average of one
sample (large enough to give a good average), and
we take a large number of samples (enough samples
to get a good confidence interval).