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Understanding Probability Models: Sample Space and Random Variables

Learn about the key components of probability models including sample space, sigma-field, events, and random variables in this comprehensive guide.

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Understanding Probability Models: Sample Space and Random Variables

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  1. ~ Three Ingredients of a “Probability Model” ~ • Sample space (of an experiment) = set of all possible outcomes • Sigma-fieldof events • Probability measure Def: A “random variable”X is a function that maps Ω to the real numbers. | 0 | {1,2,3,4,5,6} Sample space Example: Roll one die. (Discrete) Random VariableX = “Value shown” “probability mass function” pmf

  2. ~ Three Ingredients of a “Probability Model” ~ • Sample space (of an experiment) = set of all possible outcomes • Sigma-fieldof events • Probability measure Def: A “random variable”X is a function that maps Ω to the real numbers. | 0 | {1,2,3,4,5,6} Sample space “Uniform Distribution” Example: Roll one die. fair die. (Discrete) Random VariableX = “Value shown” “probability mass function” Total Area = 1 pmf X

  3. ~ Three Ingredients of a “Probability Model” ~ • Sample space (of an experiment) = set of all possible outcomes • Sigma-fieldof events • Probability measure Def: A “random variable”X is a function that maps Ω to the real numbers. | 0 | {1,2,3,4,5,6} Sample space “Uniform Distribution” Exercise Example: Roll one die. fair die. (Discrete) Random VariableX = “Value shown” “probability mass function” Total Area = 1 pmf X

  4. ~ Three Ingredients of a “Probability Model” ~ • Sample space (of an experiment) = set of all possible outcomes • Sigma-fieldof events • Probability measure Def: A “random variable”X is a function that maps Ω to the real numbers. | 0 | Die 2 Die 1 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 Example: Roll two fair dice.

  5. ~ Three Ingredients of a “Probability Model” ~ • Sample space (of an experiment) = set of all possible outcomes • Sigma-fieldof events • Probability measure Def: A “random variable”X is a function that maps Ω to the real numbers. | 0 | Die 2 Die 1 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 Example: Roll two fair dice. Events A = “Die1 > Die2” B = “Roll doubles” C = “The sum of the two dice is 6.”

  6. Def: A “random variable”X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space 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” Die 2 Die 1 B = “Roll doubles” = “Die1 = Die2” = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} B = “Y = 0”

  7. Def: A “random variable”X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} Die 2 Die 1 Probability…

  8. Def: A “random variable”X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} Die 2 Die 1 Probability…

  9. Def: A “random variable”X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} Die 2 Die 1 Probability…

  10. Def: A “random variable”X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} Die 2 Die 1 Probability…

  11. Def: A “random variable”X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} Die 2 Die 1 Probability…

  12. Def: A “random variable”X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} Die 2 Die 1 Probability…

  13. Def: A “random variable”X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} Die 2 Die 1 Probability…

  14. Def: A “random variable”X is a function that maps Ω to the real numbers. Example: Roll two fair dice. Sample space Events C = “The sum of the two dice is 6.” = “X = 6” = {(1,5), (2,4), (3,3), (4,2), (5,1)} Die 2 Die 1 Probability… ?????? Chapter 2… Total Area = 1

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