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POPULATION (of “units”)
uniform X = “Random Variable” symmetric unimodal “REAL WORLD” SYSTEM skew (positive) PROBABILTY MODEL YES Model has to be tweaked. THEORY EXPERIMENT Is there a significant difference? Random Sample Model Predictions STATISTICS How do we test them? NO Model may be adequate / useful.
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POPULATION (of “units”)
uniform “Arbitrarily large” or infinite collection of units (rocks, toasters, people, districts,…) X = “Random Variable” symmetric unimodal “REAL WORLD” SYSTEM A numerical value assigned to each unit of the population skew (positive) PROBABILTY MODEL Temp Mass Foot length Shoe size # children TV channel Alphabet Zip Code Shirt color Coin toss Pregnant? Quantitative (A = 01,…, Z = 26) Qualitative (Blue = 1, White = 2…) (Heads = 1, Tails = 0) (Yes = 1, No = 0)
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POPULATION (of “units”)
uniform skew (positive) symmetric unimodal “Arbitrarily large” or infinite collection of units (rocks, toasters, people, districts,…) X = “Random Variable” “REAL WORLD” SYSTEM A numerical value assigned to each unit of the population PROBABILTY MODEL Temp Mass Foot length Shoe size # children TV channel Alphabet Zip Code Shirt color Coin toss Pregnant? Continuous Quantitative Discrete Ordinal (A = 01,…, Z = 26) Qualitative (Blue = 1, White = 2…) Nominal (Heads = 1, Tails = 0) Binary (Yes = 1, No = 0)
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POPULATION (of “units”)
uniform skew (positive) symmetric unimodal “Arbitrarily large” or infinite collection of units (rocks, toasters, people, districts,…) X = “Random Variable” “REAL WORLD” SYSTEM A numerical value assigned to each unit of the population PROBABILTY MODEL Temp Mass Foot length Shoe size # children Coin toss Pregnant? Continuous Continuous Quantitative Discrete Discrete Qualitative (Heads = 1, Tails = 0) Binary (Yes = 1, No = 0)
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POPULATION (of “units”)
uniform skew (positive) symmetric unimodal “Arbitrarily large” or infinite collection of units (rocks, toasters, people, districts,…) X = “Random Variable” “REAL WORLD” SYSTEM A numerical value assigned to each unit of the population PROBABILTY MODEL Temp Mass Foot length Shoe size # children Continuous Continuous Quantitative Discrete Discrete “probability density function” “histogram” “probability mass function”
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~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes Sigma-field of events Probability measure Def: A “random variable” X is a function that maps Ω to the real numbers. | | Example: Roll one die. Sample space {1,2,3,4,5,6} (Discrete) Random Variable X = “Value shown” “probability mass function” pmf
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~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes Sigma-field of events Probability measure Def: A “random variable” X is a function that maps Ω to the real numbers. | | “Uniform Distribution” X Example: Roll one die. fair die. Sample space {1,2,3,4,5,6} (Discrete) Random Variable X = “Value shown” x p(x) 1 1/6 2 3 4 5 6 “probability mass function” Total Area = 1 pmf
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~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes Sigma-field of events Probability measure Def: A “random variable” X is a function that maps Ω to the real numbers. | | {1,2,3,4,5,6} “Uniform Distribution” X Example: Roll one die. fair die. Sample space Exercise (Discrete) Random Variable X = “Value shown” x p(x) 0.20 0.30 0.15 0.10 0.05 1 x p(x) 1 1/6 2 3 4 5 6 “probability mass function” Total Area = 1 pmf
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~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes Sigma-field of events Probability measure Def: A “random variable” X is a function that maps Ω to the real numbers. | | Example: Roll two fair dice. Sample space The set of all ordered pairs (i, j) such that i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6 Die 1 Die 2
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~ Three Ingredients of a “Probability Model” ~
Sample space (of an experiment) = set of all possible outcomes Sigma-field of events Probability measure Def: A “random variable” X is a function that maps Ω to the real numbers. | | Example: Roll two fair dice. Sample space The set of all ordered pairs (i, j) such that i = 1,2,3,4,5,6 and j = 1,2,3,4,5,6 Die 1 Die 2 Events A = “Die1 > Die2” B = “Roll doubles” C = “The sum of the two dice is 6.”
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Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice. Sample space Die 1 Die 2 Events A = “Die1 > Die2” = {(2,1), (3,1), (3,2), (4,1), (4,2), (4,3), (5,1), (5,2), (5,3), (5,4), (6,1), (6,2), (6,3), (6,4), (6,5)} A = “Y > 0” B = “Roll doubles” = “Die1 = Die2” = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} B = “Y = 0”
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Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Die 1 Die 2 Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} x pmf p(x) = P(X = x) 2 3 4 5 6 7 8 9 10 11 12 1 Probability…
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Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Die 1 Die 2 Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} x Outcomes pmf p(x) = P(X = x) 2 3 4 5 6 7 8 9 10 11 12 1 Probability…
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Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Die 1 Die 2 Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} x Outcomes pmf p(x) = P(X = x) 2 (1,1) 3 4 5 6 7 8 9 10 11 12 1 Probability…
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Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Die 1 Die 2 Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} x Outcomes pmf p(x) = P(X = x) 2 (1,1) 3 (1,2), (2,1) 4 5 6 7 8 9 10 11 12 1 Probability…
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Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Die 1 Die 2 Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} x Outcomes pmf p(x) = P(X = x) 2 (1,1) 3 (1,2), (2,1) 4 (1,3), (2,2), (3,1) 5 6 7 8 9 10 11 12 1 Probability…
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Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Die 1 Die 2 Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} x Outcomes pmf p(x) = P(X = x) 2 (1,1) 3 (1,2), (2,1) 4 (1,3), (2,2), (3,1) 5 (1,4), (2,3), (3,2), (4,1) 6 (1,5), (2,4), (3,3), (4,2), (5,1) 7 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 8 (2,6), (3,5), (4,4), (5,3), (6,2) 9 (3,6), (4,5), (5,4), (6,3) 10 (4,6), (5,5), (6,4) 11 (5,6), (6,5) 12 (6,6) 1 Probability…
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Example: Roll two fair dice. Sample space
Def: A “random variable” X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Die 1 Die 2 Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} x Outcomes pmf p(x) = P(X = x) 2 (1,1) 1/36 3 (1,2), (2,1) 2/36 4 (1,3), (2,2), (3,1) 3/36 5 (1,4), (2,3), (3,2), (4,1) 4/36 6 (1,5), (2,4), (3,3), (4,2), (5,1) 5/36 7 (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) 6/36 8 (2,6), (3,5), (4,4), (5,3), (6,2) 9 (3,6), (4,5), (5,4), (6,3) 10 (4,6), (5,5), (6,4) 11 (5,6), (6,5) 12 (6,6) 1 Probability…
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Def: A “random variable” X is a function that maps Ω to the real numbers.
Example: Roll two fair dice. Sample space Die 1 Die 2 Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} x pmf p(x) = P(X = x) 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 9 10 11 12 1 Probability… ?????? Chapter 2… Total Area = 1
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