**Toolbox:**

- X={\(0\;1\;2\)}
- \(P(X=0)+P(X=1)+P(X=2)\)=1
- \(P+P+P(X=2)\)=1
- \(p(X=2)\)=1-2p
- Also\(E(X)\)=\(P(X=0)\times0+P(X=1)\times1+P(X=2)\times2\)
- \(E(X)^2\)=\(P(X=0)\times0^{2}+P(X=1)\times1^{2}+P(X=2)\times2^{2}\)

E(\(x^2)\)=\(O\times\;P+1^{2}\times\;P+2^{2}(1-2p)\)=\(P+4-8p\)=\(4-7p\)

E(\(x)\)=\(O\times\;P+1\times\)P+2(1-2p)=\(P+2-4p\)=\(2-3p\)

sinceE(\(X^2\))=E(\(x\))

=\(2-3p=4-7p\)

=\(4p=2\)

\(p=\Large\frac{1}{2}\)