Example on Variance (Sheldon Ross)

Calculate \(Var(X)\) if \(X\) represents the outcome when a fair die is rolled.

Answer

Things you need to know

\(Var(X) = E[X^2] - (E[X])^2\)

Now,

\(E[X] = \frac{1}{6}(1) + \frac{1}{6}(2) + \frac{1}{6}(3) + \frac{1}{6}(4) + \frac{1}{6}(5) + \frac{1}{6}(6) \\ \ \ \ \ \ \ \ \ \ = \frac{1}{6}(\frac{6*7}{2}) \\ \ \ \ \ \ \ \ \ \ = \frac{7}{2}\)

 

\(E[X^2] = 1^2(\frac{1}{6}) + 2^2(\frac{1}{6}) + 3^2(\frac{1}{6}) + 4^2(\frac{1}{6}) + 5^2(\frac{1}{6}) + 6^2(\frac{1}{6}) \\
\ \ \ \ \ \ \ \ \ \ \ = \frac{91}{6}\)

Hence, \(Var(X) = \frac{91}{6} - (\frac{7}{2})^2 = \frac{35}{12}\)

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