One Random Variable
Random Process
The Cumulative Distribution Function
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We have already known that the probability mass function
of a discrete random variable is  X  b
The cumulative distribution function is an alternative
approach, that is  X  b
The most important thing is that the cumulative
distribution function is not limited to discrete random
variables, it applies to all types of random variables
Formal definition of random variable
Consider a random experiment with sample space S and
event class F. A random variable X is a function from the
sample space S to R with the property that the set
Ab   : X    b is in F for every b in R
The Cumulative Distribution Function
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The cumulative distribution function (cdf) of a random
variable X is defined as
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The cdf is a convenient way of specifying the probability
of all semi-infinite intervals of the real line (-∞, b]
Example 1
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From last lecture’s example we know that the number of
heads in three tosses of a fair coin takes the values of 0, 1,
2, and 3 with probabilities of 1/8, 3/8, 3/8, and 1/8
respectively
The cdf is the sum of the probabilities of the outcomes
from {0, 1, 2, 3} that are less than or equal to x
Example 2
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The waiting time X of a costumer at a taxi stand is zero if
the costumer finds a taxi parked at the stand
It is a uniformly distributed random length of time in the
interval [0, 1] hours if no taxi is found upon arrival
Assume that the probability that a taxi is at the stand
when the costumer arrives is p
The cdf can be obtained as follows
The Cumulative Distribution Function
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The cdf has the following properties:
Example 3
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Let X be the number of heads in three tosses of a fair
coin
The probability of event
can be obtained
by using property (vi)
The probability of event
can be
obtained by realizing that the cdf is continuous at
and
Example 3 (Cont’d)
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The cdf for event
getting first
By using property (vii)
can be obtained by
Types of Random Variable
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Discrete random variables: have a cdf that is a rightcontinuous staircase function of x, with jumps at a
countable set of points
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Continuous random variable: a random variable whose
cdf is continuous everywhere, and sufficiently smooth that
it can be written as an integral of some nonnegative
function
Types of Random Variable
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Random variable of mixed type: random variable with a
cdf that has jumps on a countable set of points, but also
increases continuously over ar least one interval of values
of x
where
variable, and
variable
,
is the cdf of a discrete random
is the cdf of a continuous random
The Probability Density Function
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The probability density function (pdf) is defined as
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The properties of pdf
The Probability Density Function
The Probability Density Function
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A valid pdf can be formed from any nonnegative,
piecewise continuous function that has a finite integral
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If
, the function will be normalized
Example 4
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The pdf of the uniform random variable is given by
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The cdf will be
Example 5
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The pdf of the samples of the amplitude of speech
waveform is decaying exponentially at a rate α
In general we define it as
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The constant, c can be determined by using normalization
condition as follows
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Therefore, we have
We can also find
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Pdf of Discrete Random Variable
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Remember these:
Unit step function
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The pdf for a discrete random variable is
Example 6
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Let X be the number of head in three coin tosses
The cdf of X is
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Thus, the pdf is
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We can also find several probabilities as follows
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Conditional Cdf’s and Pdf’s
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The conditional cdf of X given C is
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The conditional pdf of X given C is
The Expected Value of X
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The expected value or mean of a random variable X is
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Let Y = g(X), then the expected value of Y is
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The variance and standard deviation of the random
variable X are
The Expected Value of X
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The properties of variance
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The n-th moment of the random variable is
Some Continuous Random Variable
Some Continuous Random Variable
Some Continuous Random Variable
Some Continuous Random Variable
Some Continuous Random Variable
Transform Methods
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Remember that when we perform convolution between
two continuous signal f1 t   f2 t  , we can perform it in
another way
First we do transformation (that is, Fourier transform), so
that we have
F  f1  t   f 2  t    F1    F2  
Transform Methods
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The characteristic function of a random variable X is
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The inversion formula that represent pdf is
Example 7: Exponential Random Variable
Transform Methods
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If we subtitute
of
yields
into the formula
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When the random variables are integer-valued, the
characteristic function is called Fourier transform of the
sequence
as follows
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The inverse:
Example 8: Geometric Random Variable
Transform Methods
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The moment theorem states that the moments of X are
given by
Example 9
The Probability Generating Function
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The probability generating function of a nonnegative
integer-valued random variable N is defined by
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The pmf of N is given by
The Laplace Transform of The Pdf
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The Laplace transform of the pdf can be written as
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The moment theorem also holds
Example 10