Discrete Random Variables
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
Random Variable Defined
X( )A random variable is the assignment of numerical
values to the outcomes of experiments
Random VariablesExamples of assignments of numbers to the outcomes of
experiments.
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
The probability mass function (pmf ) for a discrete random
variable X is
PX
x( ) = P X = x .
Probability Mass Functions
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
Example of a Bernoulli pmf
Probability Mass Functions
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
Binomial pmf
Probability Mass Functions
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
Geometric pmf
Probability Mass Functions
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
Negative Binomial
(Pascal) pmf
Probability Mass Functions
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
Events occurring at random times
Probability Mass Functions
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
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
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.
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.
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
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
( )
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.
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
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.
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 .
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 .
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( )
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