4  Binomial distribution

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Let \(\mathcal Y=\set{0,\ldots,n}\).

Definition 4.1 (Binomial distribution) \(Y\in\mathcal Y\) has a binomial distribution with probability \(\theta\), denoted by \(Y\sim\distbinom(n, \theta)\), if \[ p(y\mid\theta)=\pr\rdb{Y=y\mid \theta}=\pdfbinom(y,n,\theta)=\dbinom{n}{y}\theta^y(1-\theta)^{n-y}. \]

4.1 Expected values

Theorem 4.1 Let \(Y\sim\distbinom(n,\theta)\). Then \[ \Exp\sqb{Y}=n\theta. \]

Proof. \[ \begin{split} \Exp\sqb{Y}&=\sum_{y=0}^n\dbinom{n}{y}y\theta^y(1-\theta)^{n-y}=\sum_{y=0}^n\frac{n!}{y!(n-y)!}y\theta^y(1-\theta)^{n-y}\\ &=\sum_{y=1}^nn\theta\frac{(n-1)!}{(y-1)!(n-1-(y-1))!}\theta^{y-1}(1-\theta)^{n-y}\\ &=n\theta\sum_{y=0}^{n-1}\frac{(n-1)!}{y!(n-1-y)!}\theta^y(1-\theta)^{n-1-y}\\ &=n\theta(\theta+1-\theta)^{n-1}=n\theta. \end{split} \]

4.2 Variance

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