Moment generating function

Emin Gabrielyan / Aram Gabrielyan

2010-09-22 < 2010-09-22 < 2010-09-22 < 2010-09-22 < 2010-09-22

 

 

Moment generating function. 1

1.     The logarithm of the moment generating function. 1

1.1.      The task. 1

1.2.      Proof 2

1.3.      The task. 2

1.4.      Proof 3

1.5.      References. 4

2.     Does a distribution exist for a given momentum generating function?. 4

2.1.      The task. 4

2.2.      The answer 4

3.     Moment generating function of a discrete distribution. 5

3.1.      The task. 5

3.2.      The moment generating function of a uniform distribution. 5

3.3.      The moment generating function of a degenerate distribution. 6

3.4.      The moment generating function of a discrete distribution with two possible values. 7

3.5.      The moment generating function of a distribution with multiple discrete values. 7

3.6.      The discrete distribution behind the moment generating function of this task. 8

3.7.      The answer 9

3.8.      References. 9

 

 

 

1.       The logarithm of the moment generating function

 

1.1.            The task

 

Let

 

 

be the moment generating function of X, and define

 

 

Show

 

 

 

1.2.            Proof

 

 

 

1.3.            The task

 

Let

 

 

be the moment generating function of X, and define

 

 

Show

 

 

 

1.4.            Proof

 

 

 

1.5.            References

 

http://en.wikipedia.org/wiki/Variance

 

http://en.wikipedia.org/wiki/Derivative

 

 

2.       Does a distribution exist for a given momentum generating function?

 

2.1.            The task

 

Does a distribution exist for which

 

 

If yes find it. If no, prove it.

 

 

2.2.            The answer

 

 

As

 

 

There does not exist a distribution for

 

 

 

3.       Moment generating function of a discrete distribution

 

3.1.            The task

 

For a random variable X

 

 

Find

 

 

 

3.2.            The moment generating function of a uniform distribution

 

Let us compute the moment generating function of a uniform distribution

 

By definition of the uniform probability density function:

 

 

By definition of the moment generating function:

 

 

By derivative chain rule:

 

 

Therefore:

 

 

 

3.3.            The moment generating function of a degenerate distribution

 

Let us compute the moment generating function of a degenerate distribution, i.e. a discrete distribution with only one possible value.

 

 

The moment generating function of a degenerate function is the extreme case of the uniform distribution with the upper bound infinitively approaching to the lower bound.

 

 

Therefore:

 

 

Replace

 

 

By

 

 

The expression of the limit in the equation becomes the definition of the derivative of the exponential function at the point 0.

 

 

And finally, we write the moment generating function of the degenerate distribution:

 

 

 

3.4.            The moment generating function of a discrete distribution with two possible values

 

A discrete distribution with two possible values can be represented as a weighted sum of two degenerated distributions:

 

 

Where

 

 

The moment generating function of the random variable with two possible values is:

 

 

 

3.5.            The moment generating function of a distribution with multiple discrete values

 

 

Similarly to the moment generating function with two possible values only, we can represent the moment generating function also for a discrete distribution with N possible values:

 

 

Where

 

 

The moment generating function now can be written as follows:

 

 

 

3.6.            The discrete distribution behind the moment generating function of this task

 

The moment generating function of this task is shown below:

 

 

Let us open the parenthesis:

 

 

As 1+8+24+32+16=81, the condition of the sum of probabilities is respected:

 

 

Therefore we are dealing with a random variable distribution over the 5 discrete values:

 

4, 3, 2, 1, and 0

 

With the following probabilities:

 

1/81, 8/81, 24/81, 32/81, and 16/81

 

The probability density function corresponding to this task is the following:

 

 

 

3.7.            The answer

 

The probability, that:

 

 

Is equal to 24/81 + 32/81 + 16/81 = 72/81 = 8/9

 

The answer is 8/9

 

 

3.8.            References

 

http://en.wikipedia.org/wiki/Moment-generating_function

 

The moment generating function of a uniform distribution [xls]

 

http://switzernet.com/people/aram-gabrielyan/public/100922-moment-gen-discrete-prob/

http://mirror2.switzernet.com/people/aram-gabrielyan/public/100922-moment-gen-discrete-prob/

 

The task source [pdf], [eml], [mht]

 

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