statistics

Mgf of binomial distribution

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The_Don. answered · 15/04/2021

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Answer

The MGF of a Binomial Random variable X, with parameters (n,p) is given by;

$M_{X} (t) = E (e^{t x})$

$= \sum_{x = 0}^{n} e^{t x} (\begin{matrix} n \\ x \end{matrix}) p^{x} q^{n - x}$

$= {\sum (P}_{x = 0}^{n} e^{t})^{X} (\begin{matrix} n \\ x \end{matrix}) q^{n - x}$

$R e c a l l; {(a + b)}^{n} = \sum_{0}^{n} (\begin{matrix} n \\ x \end{matrix}) a^{x} b^{n - x}$

$L e t a = P e^{t} b = q$

$M_{x} (t) = {(P e^{t} + q)}^{n}$

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