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Probability

5.1 Joint Discrete Distributions

Alex Tsun

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Agenda

Cartesian Products of Sets
Joint PMFs and Expectation
Marginal PMFs
Independence
Variance Adds for Independent RVs

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Cartesian Product of Sets

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Cartesian Product of Sets

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Cartesian Product of Sets

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Cartesian Product of Sets

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Example:weird dice Again

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Example:weird dice Again

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Example:weird dice Again

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Example:weird dice Again

X\Y	1	2	3	4
1
2
3
4

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Example:weird dice Again

X\Y	1	2	3	4
1
2
3
4

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Example:weird dice Again

X\Y	1	2	3	4
1	1/16	1/16	1/16	1/16
2	1/16	1/16	1/16	1/16
3	1/16	1/16	1/16	1/16
4	1/16	1/16	1/16	1/16

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Example:weird dice Again

X\Y	1	2	3	4
1	1/16	1/16	1/16	1/16
2	1/16	1/16	1/16	1/16
3	1/16	1/16	1/16	1/16
4	1/16	1/16	1/16	1/16

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Joint PMFs and Expectation

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Example:weird dice Again

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Example:weird dice Again

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Example:weird dice Again

U\V	1	2	3	4
1
2
3
4

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Example:weird dice Again

U\V	1	2	3	4
1
2
3
4

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Example:weird dice Again

U\V	1	2	3	4
1
2	0
3	0	0
4	0	0	0

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Example:weird dice Again

U\V	1	2	3	4
1	1/16
2	0	1/16
3	0	0	1/16
4	0	0	0	1/16

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Example:weird dice Again

U\V	1	2	3	4
1	1/16	2/16	2/16	2/16
2	0	1/16	2/16	2/16
3	0	0	1/16	2/16
4	0	0	0	1/16

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Example:weird dice Again

U\V	1	2	3	4
1	1/16	2/16	2/16	2/16
2	0	1/16	2/16	2/16
3	0	0	1/16	2/16
4	0	0	0	1/16

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Example:weird dice Again

U\V	1	2	3	4
1	1/16	2/16	2/16	2/16
2	0	1/16	2/16	2/16
3	0	0	1/16	2/16
4	0	0	0	1/16

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Example:weird dice Again

U\V	1	2	3	4
1	1/16	2/16	2/16	2/16
2	0	1/16	2/16	2/16
3	0	0	1/16	2/16
4	0	0	0	1/16

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Example:weird dice Again

U\V	1	2	3	4
1	1/16	2/16	2/16	2/16
2	0	1/16	2/16	2/16
3	0	0	1/16	2/16
4	0	0	0	1/16

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Example:weird dice Again

U\V	1	2	3	4
1	1/16	2/16	2/16	2/16
2	0	1/16	2/16	2/16
3	0	0	1/16	2/16
4	0	0	0	1/16

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Example:weird dice Again

U\V	1	2	3	4
1	1/16	2/16	2/16	2/16
2	0	1/16	2/16	2/16
3	0	0	1/16	2/16
4	0	0	0	1/16

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Marginal PMFs

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Marginal PMFs

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Marginal PMFs

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Random Picture

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Independence

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Independence

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Independence

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Variance adds for Independent RVs

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Variance adds for Independent RVs

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Variance Adds for Independent RVs (Lemma)

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Variance Adds for Independent RVs (Lemma)

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Variance Adds for Independent RVs (Lemma)

(Independence)

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Variance Adds for Independent RVs (Lemma)

(Independence)

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Variance Adds for Independent RVs (Lemma)

(Independence)

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