Step by Step Solutions

(Solution Library) Let X be a random variable. Its generating function is defined as G_X(s)=E(s^X) Let X ~ Binomial(n, p). Show that G_X(s)=[ (1-p)+ps ]^n Let Y_1,...,Y_n

Question: Let X be a random variable. Its generating function is defined as

\[{{G}_{X}}\left( s \right)=E\left( {{s}^{X}} \right)\]
  1. Let X ~ Binomial(n, p). Show that
    \[{{G}_{X}}\left( s \right)={{\left[ \left( 1-p \right)+ps \right]}^{n}}\]
  2. Let \({{Y}_{1}},...,{{Y}_{n}}\) be Bernoulli(p) be independent random variables. Show that
    \[{{G}_{{{Y}_{i}}}}\left( s \right)=\left( 1-p \right)+ps\]
  3. Show that the sum of n Bernoulli random variables is a binomial random variable.

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Deliverable: Word Document


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