Joint Probability Distribution

Gamma is used for multiple events

Expected Values

Covariation

Correlation

Single

Joint

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$E(x) = \int{x *f(x)}$
$E(x) = \int{h(x) * f(x)}$

Single

Joint

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Concept

Shows how one variable after another

Positive

Negative

Neutral

X changing doesn't change Y

Increasing X increases Y

Increasing X decreases Y

inversely proportional

proportional

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Not great for showing how strongly the data is

That is where correlation comes in

Concept

Shows how strongly related or weakly related the variables are

[-1,1]

Strong relation means

Even though you can have the same correlation the graphs can vary drastically. Anscombe 's quartet

Independence

Correlation of O doesn't imply independence
Independence however does imply Correlation= O

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