TELECOMMUNICATIONS

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TELECOMMUNICATIONS

• Dr. Hugh Blanton
• ENTC 4307/ENTC 5307

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Random Variables
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Random Variables

• Many random phenomena have outcomes that are real
  numbers,
• e.g., the voltage, v(t) at time, t, across a
  noisy resistor, number of people on a New York to
  Chicago train, etc.
• In engineering, technology, and science we are
  generally interested in numerical outcomes.
• Even when the universal set, S, in not numerical,
  we may apply a mapping to convert the outcomes to
  real numbers.

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Definition of a Random Variable

• A random variable is a number labeling the
  outcomes of a probabilistic experiments.
• X can be considered to be a function that maps
  all the elements in S into points on the real
  line or some parts thereof.

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Conditions The mapping is single-valued. The set
X ? x is an event. This is the set of random
variable X taking values equal or less than x in
a trial chance experiment, E.
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Basic Definitions

• Discrete Random Variable A random variable that
  has a countable number of elements in the range.
• Continuous Random Variable A random variable
  that has an uncountably infinite number of
  elements in the range.

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Random Variables

• The mapping (function) that assigns a number to
  each outcome is called a random variable.
• If the random variable is denoted by X, then the
  distribution function F(xo) is defined by

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Example 1
Suppose you match coins with a friend, winning 1
if two coins match and losing 1 if the coins do
not match. Example 1 SHH, HT, TH, TT
s1 s2 s3 s4 Random Variable
X(s1) X(s4) 1 X(s2 ) X(s3)
-1 Thus,
X 1 -1 -1 1
S HH HT TH TT
? Single-valued mapping
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In this case, a random variable takes on only a
finite number of values (1, -1), satisfying
property c. If we let x 0.6, then X ? 0.6, if s
HT or TH, i.e., the event HT, TH. Thus x
0.6 determines an event. Let x -10, the X ?
-10 Ø Let x gt 1, then X ? x S Thus, for
every x, we have an event and b is satisfied.
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Basic Definitions

• Discrete Random Variable A random variable that
  has a countable number of elements in the range.
• Continuous Random Variable A random variable
  that has an uncountably infinite number of
  elements in the range.
• Probability Assignment There are two standard
  forms for probability assignment either using
  Cumulative Distribution Function (CDF) or
  Probability Distribution Function (PDF).

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Cumulative Distribution Function (CDF)
Let X a random variable with a particular
value, x, then, FX(x) PrX ? x Thus, the
CDF is the probability of event X ? x, i.e.,
the random variable, X, takes on a value equal to
or less than x.
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Example 2
Experiment Observing the parity bit in a word
in computer memory. Bit ON ? X
1 Bit OFF ? X 0 The OFF state has a
probability q and thus the ON state has a
probability of (1-q). Sample space, S OFF,
ON Plot FX(x)
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Example 2
FX(x)
Prob. of event X1
Prob. of event X0
q
q
x
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Example 3
Determine CDF for a single toss of a die.
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Example 3
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1
FX(x)
1/6
1
6
x
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Example 4
A random variable has a PDF given by FX(x)
0 -? lt x ? 0 1-e-2x 0 lt x ? ? Find the
probability that X gt 0.5. Find the probability
that X ? 0.25 Find the probability that 0.3 ? X ?
0.7
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Example 4
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FX(x)
1
x
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Example 5
A random variable has PDF given by FX(x)
A(1-e-(x-1)) 1lt x lt ? 0 - ? lt x ? 1 Find
A for a valid CDF FX(x) ? Pr2 lt X lt ?
? Pr1 lt X ? 3 ?
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Example 5

• Since FX(?) 1,
• ? A 1 e-? ? A 1
• FX(2) 1 e-1 0.6321
• Pr2 lt X lt ? FX(?) - FX(2) 1 - 0.6321
  0.3679
• Pr1 lt X ? 3 FX(3) - FX(1)
• (1 e-2) - (1 e0) 0.8647

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CDF or Discrete Random Variable
A discrete random variable , X, taking on one of
the countable set of possible values x1, x2, ?
with probability PrX xk, ? k?1,N forming a
stair-step CDF with amplitude of each step being
PrX xk, k 1, 2, ?. Thus, where, Or
more compactly,
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Example 6
A bus arrives at random in (0, T, i.e., 0 lt t ?
T. Let X be a random variable representing time
of arrival, then clearly, FX(t) 0 for t ?
0 impossible event FX(T) 1 certain event Bus
is uniformly likely to come at any time within
(0,T. Then A continuous random variable has a
continuous CDF.
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Probability Density Function (PDF)
A PDF is defined as Properties of PDF
If fX(x) exists, then (1)
i.e., CDF (2)
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(3) If a -? and b ?, then (4) since
CDF is non-decreasing From (2), the probability
that X takes on values between x and x ?x is
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Generalization

• For discrete random variables, the PDF has a
  general form of
• Example 8 For a random variable, X, we have
• Find A so that this function is a valid PDF.
• Find Pr1/2 ? x 1.

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Example 8
(a)
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Example 8 (cont.)
(b)