Answer to Question #136612 in Statistics and Probability for Elaine Mabote

Question #136612
Let X~Bin(n,p).Find E(e^tx) where t is a constant
1
Expert's answer
2020-10-05T17:39:28-0400

Using independence, EetΣXi=EΠ\Pi etXi=Π\PiEetXi=(pet+(1-p))n,

where the Xi are independent Bernoulli random variables. Equivalently

k=0netk(kn)pk(1p)nk=k=0n(kn)(pet)k(1p)nk=(pet+(1p)n)\sum^n_{k=0}e^{tk}(^n_k)p^k(1-p)^{n-k}=\sum^n_{k=0}(^n_k)(pe^{t})^k(1-p)^{n-k}=(pe^t+(1-p)^n) by the Binomial formula.


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