econometrics ch 2 - Random Variables and Probability...

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Unformatted text preview: Random Variables and Probability Distributions Expected Values, Mean and Variance Two Random Variables Some Common Distributions Random Sampling and the Distribution of the Sample Average Law of Large Numbers and the Central Limit Theorem Introduction to Econometrics Chapter 2: Review of Probability Geo rey Williams gwilliams@econ.rutgers.edu September 8, 2010 Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 2: Review of Probab Random Variables and Probability Distributions Expected Values, Mean and Variance Two Random Variables Some Common Distributions Random Sampling and the Distribution of the Sample Average Law of Large Numbers and the Central Limit Theorem Probability and Randomness Probability is the mathematical study and analysis of random processes. Randomness is given a nice (informal) de nition in the book: a process is random if there is something we don't know at the beginning of the process that is revealed by the end. Examples: Coin ip Spinning a roulette wheel Applying for colleges, or jobs How the economy will develop over the next 12 months Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 2: Review of Probab Random Variables and Probability Distributions Expected Values, Mean and Variance Two Random Variables Some Common Distributions Random Sampling and the Distribution of the Sample Average Law of Large Numbers and the Central Limit Theorem Analyzing Random Processes To understand a random process, one of the rst steps is to look at outcomes , the multiple, mutually exclusive, possible results of the process. Ex: For a coin ip, there are two outcomes: Heads or Tails Every random process has a sample space , which is the full set of possible outcomes. Examples: For a coin ip, the sample space would be f heads ; tails g For a dice roll, the sample space would be f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g For a football game, the sample space would be every possible combination of scores: f ( ; ) ; ( ; 1 ) ;:::; ( ; 200 ) ; ( 1 ; 1 ) ; ( 1 ; 2 ) ;:::; ( 200 ; 198 ) ; ( 200 ; 199 ) ; ( 20 Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 2: Review of Probab Random Variables and Probability Distributions Expected Values, Mean and Variance Two Random Variables Some Common Distributions Random Sampling and the Distribution of the Sample Average Law of Large Numbers and the Central Limit Theorem Events We can group outcomes together, as events . Examples: The dice rolls can be grouped as the two events f greater than 3 ; less than or equal to 3 g The sports events can be grouped as the three events: f we win ; we tie ; we lose g Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 2: Review of Probab Random Variables and Probability Distributions Expected Values, Mean and Variance Two Random Variables Some Common Distributions Random Sampling and the Distribution of the Sample Average Law of Large Numbers and the Central Limit Theorem...
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econometrics ch 2 - Random Variables and Probability...

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