Generating Functions for Uniform-based Distributions

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Section 6.5 Generating Functions for Uniform-based Distributions

Moment Generating Functions 5.5.1 can be derived for each of the distributions in this chapter.

Theorem 6.5.1. Moment Generating Function for Discrete Uniform.

For a discrete random variable X on the space

R

= {1, 2, ..., n},

M (t) = \frac{1}{n} \cdot [e^{t} + e^{2 t} + . . . e^{n t}]

Proof.

Presuming

R

= {1, 2, ..., n},

M (t) = \sum_{x = 1}^{n} e^{t x} / n = \frac{1}{n} \cdot [e^{t} + e^{2 t} + . . . e^{n t}]

Theorem 6.5.2. Moment Generating Function for Continuous Uniform.

For a continuous random variable X on the space

R = [a, b],

M (t) = \frac{e^{b x} - e^{a x}}{b - a}

Proof.

Presuming

R

= [a,b],

M (t) = \int_{a}^{b} e^{t x} \frac{1}{b - a} d x = \frac{1}{b - a} \frac{1}{t} e^{t x} |_{a}^{b} = \frac{e^{b x} - e^{a x}}{b - a}

Theorem 6.5.3. Moment Generating Function for Hypergeometric.

For a hypergeometric random variable over the space

R

= {0, 1, ..., min(

r, n_{1}

)}.

M (t) = \sum (e^{t x} \frac{(\binom{n_{1}}{x}) (\binom{n - n_{1}}{r - x})}{(\binom{n}{r})}) .

There is not a single easy formula that captures this summation nicely but with particular values for the parameters perhaps something could be simplified.

Proof.

\begin{aligned} M (t) & = \sum_{x = 0}^{n_{1}} e^{t x} \frac{(\binom{n_{1}}{x}) (\binom{n - n_{1}}{r - x})}{(\binom{n}{r})} \\ = a mess... \end{aligned}

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