What is the Expected Value?
Expected value is the average value expected after sampling a distribution an infinite number of times. The expected value is found by summing up the product of the probability of each possible outcome and the outcome's value.
$$E(x)=\underset{n=1}{\overset{N}{\sum}}x_nP(x_n)=\int x p(x)\; dx$$
$(x_n)$
$(x_n)$ represents a possible event.
$P(x_n)$
$P(x_n)$ is the likelihood of the possible event occurring.
Example - Death Valley Daily High Temperatures, July 2013
- July 1 - 114
- July 2 - 115
- July 3 - 115
- July 4 - 115
- July 5 - 115
- July 6 - 116
- July 7 - 116
- July 8 - 116
- July 9 - 116
- July 10 - 116
- July 11 - 116
- July 12 - 117
- July 13 - 117
- July 14 - 117
- July 15 - 117
- July 16 - 117
- July 17 - 117
- July 18 - 117
- July 19 - 117
- July 20 - 117
- July 21 - 117
- July 22 - 117
- July 23 - 117
- July 24 - 117
- July 25 - 117
- July 26 - 117
- July 27 - 117
- July 28 - 117
- July 29 - 117
- July 30 - 117
- July 31 - 117
Mean: 116.5 | Median: 117 | Mode: 117
$$Expected\; Value = 114*(\frac{1}{31}) + 115*(\frac{4}{31}) + 116*(\frac{6}{31}) + 117*(\frac{20}{31}) = 116.5$$
