Download Expected Value, Standard Deviation, and Variance

Expected Value, Standard
Deviation, and Variance
Christian, Harrison, Michael R
Expected Value
Definition:
- “The value one would expect to find if one performed the
experiment ∞ times”
Formula:
- E(x) = ∫Ixf(x)dx
Variance & Standard Deviation
Variance: The variance ( var(x) ) of a random variable X on
interval I roughly measures the “deviation” (how far each
f(x) is away from E(x) )
Var(x) is given by the equation:
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var(x)=∫I[x-E(x)]2f(x)dx
Variance and Standard Deviation (Cont.)
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The standard deviation is given by:
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σ = √(var(x)
If:
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●
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σ = 0 then all outcomes are identical
σ = a small number, the points are clustered closely around the mean
σ = a large number, the points are farther from the mean
Normal Distribution
-
Most phenomena have probability distributions that are normal!
Mean = µ
Normal Distribution Continued…
To find what percentage lies within a
certain range, we first find our z-score with
this equation:
Finally, if we’re trying to find the percentage of
responses that lie in between two z-score, we simply
subtract the smaller from the bigger...
Then, we use this chart to find the
percentage of responses that lie under
each z-score.