statistics

Mean of binomial distribution

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The_Don. answered · 15/04/2021

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Answer

Mean of Binomial Distribution Proof

The MGF of Binomial Distribution is given by; $M_{X} (t) = {(P e^{t} + q)}^{n}$

To get the Mean using the MGF Technique, we use the first derivative;

$M_{X}^{I} (t) = n {(P e^{t} + q)}^{n - 1} \times P e^{t} n P e^{t} {(P e^{t} + q)}^{n - 1}$

$R e p l a c e t h e t w i t h 0$

$M_{X}^{I} (0) = n {(P e^{0} + q)}^{n - 1} \times P e^{0}$

$= n p {(p + q)}^{n - 1} R e m e m b e r t h a t p + q = 1$

$A n d 1 r a i s e d t o a n y p o w e r = 1$

=np

$T h u s t h e M e a n o f B i n o m i a l D i s t r i b u t i o n = n p$

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