discrete random variables - university of...
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Discrete Random Variables
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Randomness
• The word random effectively means
unpredictable
• In engineering practice we may treat some
signals as random to simplify the analysis
even though they may not actually be
random
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Random Variable Defined
X( )A random variable is the assignment of numerical
values to the outcomes of experiments
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Random VariablesExamples of assignments of numbers to the outcomes of
experiments.
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Discrete-Value vs Continuous-
Value Random Variables• A discrete-value (DV) random variable has a set
of distinct values separated by values that cannot
occur
• A random variable associated with the outcomes
of coin flips, card draws, dice tosses, etc... would
be DV random variable
• A continuous-value (CV) random variable may
take on any value in a continuum of values which
may be finite or infinite in size
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The probability mass function (pmf ) for a discrete random
variable X is
PX
x( ) = P X = x .
Probability Mass Functions
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A DV random variable X is a Bernoulli random variable if it
takes on only two values 0 and 1 and its pmf is
PX
x( ) =
1 p , x = 0
p , x = 1
0 , otherwise
and 0 < p < 1.
Probability Mass Functions
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Example of a Bernoulli pmf
Probability Mass Functions
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If we perform n trials of an experiment whose outcome is
Bernoulli distributed and if X represents the total number of 1’s
that occur in those n trials, then X is said to be a Binomial random
variable and its pmf is
PX
x( ) =
n
xp
x 1 p( )n x
, x 0,1,2, ,n{ }
0 , otherwise
Probability Mass Functions
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Binomial pmf
Probability Mass Functions
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If we perform Bernoulli trials until a 1 (success) occurs and the
probability of a 1 on any single trial is p, the probability that the
first success will occur on the kth trial is p 1 p( )k 1
. A DV random
variable X is said to be a Geometric random variable if its pmf is
PX
x( ) =p 1 p( )
x 1
, x 1,2,3,...{ }0 , otherwise
Probability Mass Functions
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Geometric pmf
Probability Mass Functions
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If we perform Bernoulli trials until the rth 1 occurs and the
probability of a 1 on any single trial is p, the probability that the
rth success will occur on the kth trial is
P rth success on kth trial( ) =k 1
r 1pr 1 p( )
k r
.
A DV random variable Y is said to be a negative - Binomial
or Pascal random variable with parameters r and p if its pmf is
PY
y( ) =
y 1
r 1pr 1 p( )
y r
, y r,r +1, ,{ }
0 , otherwise
Probability Mass Functions
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Negative Binomial
(Pascal) pmf
Probability Mass Functions
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Suppose we randomly place n points in the time interval 0 t < T
with each point being equally likely to fall anywhere in that range.
The probability that k of them fall inside an interval of length t < T
inside that range is
P k inside t =n
kpk 1 p( )
n k
=n!
k! n k( )!pk 1 p( )
n k
where p = t / T is the probability that any single point falls within
t . Further, suppose that as n , n / T = , a constant. If
is constant and n that implies that T and p 0. Then
is the average number of points per unit time, over all time.
Probability Mass Functions
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Events occurring at random times
Probability Mass Functions
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It can be shown that
P k inside t =
k
k!limn
1n
n
=e
=
k
k!e
where = t. A DV random variable is a Poisson random
variable with parameter if its pmf is
PX
x( ) =
x
x!e , x 0,1,2, ,{ }
0 , otherwise
Probability Mass Functions
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Cumulative Distribution
Functions
The cumulative distribution function (CDF) is defined by
FX
x( ) = P X x .
For example, the CDF for tossing a single die is
FX
x( ) = 1/ 6( )u x 1( ) + u x 2( ) + u x 3( )+ u x 4( ) + u x 5( ) + u x 6( )
where u x( )1 , x 0
0 , x < 0
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Functions of a Random Variable
Consider a transformation from a DV random variable X
to another DV random variable Y through Y = g X( ) . If the
function g is invertible, then X = g 1Y( ) and the pmf for Y is
PY
y( ) = PX
g 1y( )( ) where P
Xx( ) is the pmf for X.
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Functions of a Random Variable
If the function g is not invertible the pmf and pdf of Y can be found
by finding the probability of each value of Y . Each value of X with
non-zero probability causes a non-zero probability for the
corresponding value of Y . So, for the ith value of Y ,
P Y = yi
= P X = xi,1
+ P X = xi,2
+
+ P X = xi,n
= P X = xi,k
k=1
n
The function to the right is an
example of a non-invertible
function.
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Expectation and Moments
Imagine an experiment with M possible distinct outcomes
performed N times. The average of those N outcomes is
X =1
Nn
ix
i
i=1
M
where xi is the ith distinct value of X and n
i
is the number of times that value occurred. Then
X =1
Nn
ix
i
i=1
M
=n
i
Nx
i
i=1
M
= rix
i
i=1
M
The expected value of X is
E X = limN
ni
Nx
i
i=1
M
= limN
rix
i
i=1
M
= P X = xi
xi
i=1
M
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Expectation and Moments
Three common measures are used in statistics to indicate
an "average" of a random variable are the mean, the
mode and the median. The mean is the sum of the values
divided by the number of values X =1
Nn
ix
i
i=1
M
.
The mode is the value that occurs most often.
PX
xmode
( ) PX
x( ) for all x.
The median is the value for which an equal number
of values fall above and below.
PX
X > xmedian
( ) = PX
X < xmedian
( )
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Expectation and Moments
The first moment of a random variable is its expected value
E X = xiP X = x
i
i=1
M
The second moment of a random variable is its mean-squared
value (which is the mean of its square, not the square of its
mean).
E X2
= xi
2 P X = xi
i=1
M
The name "moment" comes from the fact that it is mathematically
the same as a moment in classical mechanics.
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Expectation and Moments
The nth moment of a random variable is defined by
E Xn
= xi
n P X = xi
i=1
M
The expected value of a function g of a random variable is
E g X( ) = g X( )P X = xi
i=1
M
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Expectation and Moments
A central moment of a random variable is the moment of
that random variable after its expected value is subtracted.
E X E X( )n
= xi
E X( )n
P X = xi
i=1
M
The first central moment is always zero. The second central
moment (for real-valued random variables) is the variance,
X
2= E X E X( )
2
= xi
E X( )2
P X = xi
i=1
M
The variance of X can also be written as Var X . The positive
square root of the variance is the standard deviation.
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Expectation and Moments
Properties of expectation
E a = a , E aX = a E X , E Xn
n
= E Xn
n
where a is a constant. These properties can be use to prove
the handy relationship,
X
2= E X
2 E2X
The variance of a random variable is the mean of its square
minus the square of its mean. Another handy relation is
Var aX + b = a2 Var X .
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Conditional Probability Mass
Functions
The concept of conditional probability can be extended to a
conditional probability mass function defined by
PX |A
x( ) =
PX
x( )P A
, x A
0 , otherwise
where A is the condition that affects the probability of X .
Similarly the conditional expected value of X is
E X | A = x PX |A
x( )x B
and the conditional cumulative
distribution function for X is FX |A
x( ) = P X x | A .
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Conditional Probability
Let A be A = X a{ } where a is a constant.
Then FX |A
x( ) = P X x | X a =P X x( ) X a( )
P X a.
If a x then P X x( ) X a( ) = P X a and
FX |A
x( ) = P X x | X a =P X a
P X a= 1.
If a x then P X x( ) X a( ) = P X x and
FX |A
x( ) = P X x | X a =P X x
P X a=
FX
x( )F
Xa( )