Exam 5: Probability Distributions and Data Modeling

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The variance, Var[X], of a discrete random variable X as a weighted average of the squared deviations from the expected value: $$\operatorname { Var } [ X ] = \sum _ { j = 1 } ^ { \infty } \left( x _ { j } - E [ X ] \right) ^ { 2 f } \left( x _ { j } \right)$$ $\begin{array}{|l|l|l|l|l|l|}\hline\text { Outcome, } & \text { Probability, } \mathrm{f}(\mathrm{x}) & \mathrm{xf}(\mathrm{x}) & (\mathrm{x}-\mathrm{E}[\mathrm{X}]) & (\mathrm{x}-\mathrm{E}[\mathrm{X}])^{2} & (\mathrm{x}-\mathrm{E}[\mathrm{X}])^{2} \mathrm{f}(\mathrm{x}) \\\hline 2 & 1 / 16 & 1 / 8 & -3 & 9 & 9 / 16 \\\hline 3 & 1 / 8 & 3 / 8 & -2 & 4 & 1 / 2 \\\hline 4 & 3 / 16 & 3 / 4 & -1 & 1 & 3 / 16 \\\hline 5 & 1 / 4 & 5 / 4 & 0 & 0 & 0 \\\hline 6 & 3 / 16 & 9 / 8 & 1 & 1 & 3 / 16 \\\hline 7 & 1 / 8 & 7 / 8 & 2 & 4 & 1 / 2 \\\hline 8 & 1 / 16 & 1 / 2 & 3 & 9 & 9 / 16 \\\hline & \text { Expected Value } & 5 & & \text { Variance } & 21 / 2 \\\hline\end{array}$