Characteristic equation of binomial random variable

  • #1
sinned4789
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Homework Statement


find the characteristic equation of a binomial variable with pmf p(x) =[tex]\frac{n!}{(n-k)!k!}[/tex]*p[tex]^{k}[/tex]*(1-p)[tex]^{n-k}[/tex]

Homework Equations


characteristic equation
I(t) = [tex]\sum[/tex]p(x)*e[tex]^{tk}[/tex]

The Attempt at a Solution


I(t) = [tex]\sum[/tex][tex]\frac{n!}{(n-k)!k!}[/tex]*(p[tex]^{k}[/tex]*(1-p)[tex]^{-k}[/tex]*e[tex]^{tk}[/tex])*(1-p)[tex]^{n}[/tex]

i am stuck on this series because i don't know what to do with the combination term.
 
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  • #2
In the definition

[tex]E(tX) = \sum_{k-0}^{n}\binom n k e^{tk}p^k(1-p)^{n-k}[/tex]

group the etk with the pk and then think about what the binomial expansion of (a + b)n looks like.
 

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