Ch 4: Mathematical Expectation

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Last updated 4:59 AM on 6/24/26
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16 Terms

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What is Mathematical Expectation?

The theoretical average a random variable can take over a large number of repeated events. Synonymous with Expected Value.

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Mean of random variable E(X)

-also denoted by μx

-sum of all x * f(x)

-relate it back to finding sample mean

<p>-also denoted by μ<sub>x</sub></p><p>-sum of all x * f(x)</p><p>-relate it back to finding sample mean</p>
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Mean of function E[g(X)]

-sum of g(x) * f(x)

<p>-sum of g(x) * f(x)</p>
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Mean of Joint Probability Distribution E[g(X, Y)]

-sum of xy * f(x, y)

<p>-sum of xy * f(x, y)</p>
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Variance of Random Variable (σ2)

V(X) = E(X2)-(μ)2

<p>V(X) = E(X<sup>2</sup>)-(μ)<sup>2</sup></p>
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Covariance Function

-determines the linear relationship between two Random Variables (positive/negative)

-E(XY) - μxμy

<p>-determines the linear relationship between two Random Variables (positive/negative)</p><p>-E(XY) - μ<sub>x</sub>μ<sub>y</sub></p>
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Correlation Coefficient

-shows the direction AND strength of two Random Variables

-unitless version of Covariance

<p>-shows the direction AND strength of two Random Variables</p><p>-unitless version of Covariance</p>
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Theorem 4.5

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Theorem 4.6

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Theorem 4.7

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Theorem 4.8

X and Y MUST be independent

<p>X and Y MUST be independent</p>
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Theorem 4.9

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Corollary 4.9

If X and Y are INDEPENDENT

<p>If X and Y are INDEPENDENT</p>
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Corollary 4.11

Multiple independent variables

<p>Multiple independent variables</p>
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Chebyshev’s Theorem (Probability Distributions)

-recall Chebyshev’s Theorem from Ch 1

<p>-recall Chebyshev’s Theorem from Ch 1</p>
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Importance of Mathematical Expectation

Knowing parameters (like mean, variance, standard deviation, covariance…) is fundamental for understanding a system’s general nature. We get an idea of central tendency, variability, and trends