An Investment Has the Following Range of Outcomes and Probabilities

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\begin{array}{c}\begin{array}{c}\underline{ ext {Outcomes (Percent)}}\5 \% \7 \% \12 \%\end{array}\begin{array}{c}\underline{ ext {Probabilities of Outcomes}}\.25 \.50 \.25\end{array}\end{array}

Calculate the expected value and the standard deviation (round to two places after the decimal point where necessary).

\begin{array} { r r r l } \underline { \mathrm { K } _ { \mathrm { i } } } & \underline { \mathrm { P } _ { \mathrm { i } } } & \underline { \sum \mathrm { K } _ { \mathrm { i } } \mathrm { P } _ { \mathrm { i } } } \5 \% & .25 & 1.25 & \7 \% & .50 & 3.50 & \12 \% & .25 & 3.00 & \& & 7.75 & = \Sigma \mathrm { K } _ { \mathrm { i } } \mathrm { P } _ { \mathrm { I } }\end{array}

ext { Standard deviation } = \sigma = \sqrt { \sum \left( \mathrm { K } _ { \mathrm { i } } - \mathrm { K } _ { \mathrm { i } } ight) ^ { 2 } \mathrm { P } _ { \mathrm { i } } }

\begin{array} { r r r r r r } \underline { \mathrm { K } } _ { \mathrm { i } } & \underline { \mathrm { Bar } ^ { 2 } - \mathrm { K } _ { \mathrm { i } } } & \underline { \mathrm { P } _ { \mathrm { i } } } & \left( \mathrm { K } _ { \mathrm { i } } - \mathrm { Bar } - \mathrm { K } _ { \mathrm { i } } ight) & \left( \mathrm { K } _ { \mathrm { i } } - \mathrm { Bar } - \mathrm { K } _ { \mathrm { i } } ight) ^ { 2 } & \left( \mathrm {~K} _ { \mathrm { i } } - \mathrm { Bar } - \mathrm { K } _ { \mathrm { i } } ight) ^ { 2 } \mathrm { P } _ { \mathrm { i } } \5 \% & 7.75 \% & .25 & - 2.75 \% & 7.56 \% & 1.89 \% \7 \% & 7.75 \% & .50 & - .75 \% & 0.56 \% & 0.28 \% \12 \% & 7.75 \% & .25 & 4.25 \% & 18.06 \% & \underline { 4.52 \% } \& & & & & 6.69 \%\end{array}

ext { Standard deviation } = \sigma = \sqrt { 6.69 } = 2.59

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