eci 114 moments

7/25/2019 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. =

eci 114 moments

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