Emin Gabrielyan / Aram Gabrielyan

2010-09-22

The task

For a random variable X

Find

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:

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

The moment generating function is the extreme case of a uniform distribution:

Therefore:

A discrete distribution with two possible values can be represented as follows

Where

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

Similarly to the moment generating function with two values, for a discrete distribution with N possible values:

Where

The moment generating function is written as follows:

The moment generating function of the task is shown below:

Let us open the parenthesis:

As 1+8+24+32+16=81

Then 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, that:

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

The answer is 8/9

The moment generating function of a uniform distribution [xls]

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