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Binomial Random Variable

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Probability

Let XX be a RV. The number of successes XX in nn independent Bernoulli trials with probability pp of success. This is called a binomial RV with parameters (n,p)(n,p) with pmf p(k)=(nk)pk(1āˆ’p)nāˆ’k,Ā k∈X={0,1,⋯ ,n}p(k)={n\choose k}p^{k}(1-p)^{n-k},\ k\in\mathcal{X}=\{0,1,\cdots,n\}For RV X∼\mboxBinomial(n,p)X\sim \mbox{Binomial}(n,p) E[X]=np\mboxVar(X)=np(1āˆ’p)\begin{align*} &E[X]=np \\ &\mbox{Var}(X)=np(1-p) \end{align*}

The number of successes XX in nn independent Bernoulli trials with probability pp