Let X be binomial with parameters n and p. Evaluate E{X(X − 1)} from first principles (i.e. the definition of expectation), and hence, derive var(X). You may assume linearity of expectation, and that E(X) = np.
Let \(X\) be binomial with parameters \(n\) and \(p\).
\(X \sim \operatorname{Bin}(n, p)\)
\(E(X)=n p, \operatorname{Var}(X)=n p(1-p)\)
\(E\left(X^{2}\right)-(E(X))^{2}=n p(1-p)\)
\(E\left(X^{2}\right)=n p(1-p)+(E(X))^{2}=n p(1-p)+n^{2} p^{2}\)
\(E\left(X^{3}\right)=n(n-1)(n-2) p^{3}+3 n(n-1) p^{2}+n p\)
To evaluate,
\(E(X(X-1)(X-2))=E\left(X^{3}-3 X^{2}+2 X\right)\)
\(E(X(X-1)(X-2))=E\left(X^{3}\right)-3 E\left(X^{2}\right)+2 E(X)\)
\(E(X(X-1)(X-2))=n(n-1)(n-2) p^{3}+3 n(n-1) p^{2}+n p-3 n p(1-p)-3 n^{2} p^{2}+2 n p\)
\(E(X(X-1)(X-2))=n(n-1)(n-2) p^{3}+\left(3 n(n-1)-3 n^{2}\right) p^{2}+3 n p-3 n p(1-p)\)
\(E(X(X-1)(X-2))=n(n-1)(n-2) p^{3}+\left(3 n^{2}-3 n-3 n^{2}\right) p^{2}+3 n p-3 n p+3 n p^{2}\)
\(E(X(X-1)(X-2))=n(n-1)(n-2) p^{3}-3 n p^{2}+3 n p^{2}\)
\(\boldsymbol{E}(\boldsymbol{X}(\boldsymbol{X}-1)(\boldsymbol{X}-2))=n(\boldsymbol{n}-1)(\boldsymbol{n}-2) \boldsymbol{p}^{3}\)
Let X be binomial with parameters n and p. Evaluate E{X(X − 1)} from first principles (i.e. the definition of expectation), and hence, derive var(X). You may assume linearity of expectation, and that E(X) = np.
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