eci 114 moments
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ECI 114:
PROBABILISTIC SYSTEMS ANALYSIS
MOMENTS AND MOMENTGENERATING FUNCTIONS
INSTRUCTOR: Kaveh Zamani
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Moments
Moment Generating FunctionsCharacteristic functions (not in the exam)
Lecture 12
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Moments
The rth moment of a random variable X about the mean is called the rth central moment, and defined as:
= Where r=0, 1, 2, . It follows that
= 1;
= 0;and
= (or the second moment around the mean is variance) = (); (discrete variable) =
; (continious variable)
The rth moment of RV X about the origin, also called the rth
raw moment is defined as: =
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Mean, Variance, Skewness4
Mean "" is the first moment of values measured about theorigin (first raw moment).
Variance "" is the second moment of values measured aboutthe mean (second central moment).
Skewness "" is the third moment of values measured aboutthe mean (third central moment). Kurtosis (flatness or peakedness) is the fourth moment of
values measured about the mean (fourth central moment).Kurtosis is a measure of whether the distribution is tall and
skinny or short and squat, compared to the normal distribution
of the same variance. (Kurtosis of normal distribution is zero)
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Skewness and Kurtosis5
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Properties of expectation and Variance6
= = + = +
+ = + () = 0 = + = =
= + = +
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Moment Generating Function7
In probability theory, the moment-generating function of a
random variable is an alternative specification of itsprobability distribution. Therefore, it provides an
alternative route to analytical results compared with
working directly with probability density functions or
cumulative distribution functions.
If X is a RV, then its moment generating function is:
= () = ; discrete variable = ; continuous variable
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Example8
Assume X is a exponential random variable:
= > 0
=
=
()
= 1 1
= ; < 1 Only when t < 1. Otherwise the integral diverges and the
moment generating function does not exist. Have in mind
that moment generating function is not always exist and
only meaningful when the integral (or the sum) converges.
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Name origin9
Where the name comes from? Writing its Taylor expansion in
place of and exchanging the sum and the integral. = = (1 + + 2! +
3! + )
= 1 + +
2 +
3! + = ; First raw moment
= ; Second raw moment = ; Third raw moment
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Theorems on Moment Generating Functions10
If is the moment generating function of the randomvariable X and , ( 0) are constants and = , themoment generating function of Y is
=
If X and Y are independent random variables having moment
generating functions and respectively, then: = Uniqueness theorem: Suppose X and Y are RVs with and respectively. Then X and Y have the same probability
distributions if and only if:
= ()
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Example11
Compute the moment generating function of a uniform
random variable on [0, 1].
= = = ( 1)
Compute the moment generating function for a Poisson()
random variable.
= ;
=
.
!=
!
= = ()
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Characteristic Function (Not in the exam)12
In probability theory and statistics, the characteristic
function of a random variable completely defines itsprobability distribution. If a RV admits a probability
density function, then the characteristic function is the
inverse Fourier transform of the probability density function.Thus it provides the basis of an alternative route to
analytical results compared with working directly with
probability density functions or cumulative distributionfunctions
The characteristic function always exists unlike the moment-
generating function.
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Characteristic Function13
The expected value of the function () is called thecharacteristic function for the probability distribution p(x),where is parameter that can have any real value and
= 1. That is to say, the characteristic function of p(x) is: =
It has all properties of Moment Generating function,
however it always exists, as = 1, and we alwaysintegrating a bounded integral. =
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