05a STAT discrete random variables 1-SHORT SLIDES

05a STAT discrete random variables 1-SHORT SLIDES -...

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1 DISCRETE RANDOM VARIABLES Dr. V.R. Bencivenga Economic Statistics Economics 329 DISCRETE RANDOM VARIABLES PART 1 Outline Random variables Mathematical expectation Linear transformations Joint probability distributions Linear combinations Specific probability models: Binomial Hypergeometric Poisson Applications Profits Stocks and bonds Portfolio of stocks. Technology race between firms. Number of defectives Committees Arrivals
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2 DISCRETE RANDOM VARIABLES Outline Objectives Random variables Mathematical models of random quantities! Mathematical expectation How to “describe” probability distributions Linear transformations Joint distributions Linear combinations Specific probability models: Binomial Hypergeometric Useful classes of problems (PART 3) Poisson Setting up probability models of DGP’s, and using these models to obtain probabilities and expected outcomes PART 2
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3 DISCRETE RANDOM VARIABLES Introduction From probability theory, we have the concept of a random trial as a data-generating process (DGP). Toss two dice (repeatedly ) Survey a TV viewer (many viewers) We learned how to work with events defined on the sample space generated by the DGP. Now we learn how to work with quantitative (numerical) information about the basic outcomes, using a mathematical object called a random variable .
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4 DISCRETE RANDOM VARIABLES RANDOM VARIABLES
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5 DISCRETE RANDOM VARIABLES A random variable X is a mapping from the sample space to the real line. X provides a numerical measure of a characteristic of basic outcomes. When basic outcome e 1 occurs, the random variable takes on the value X(e 1 ). When basic outcome e 2 occurs, the random variable takes on the value X(e 2 ). And so forth! x 1 x 2 x 3 real line The mapping may be such that each basic outcome leads to a different point on the real line (a different numerical value), as illustrated above. X(e) e 2 e 1 S e 3
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6 DISCRETE RANDOM VARIABLES Alternatively, the mapping may be such that each basic outcome in a composite event leads to the same numerical value. x 1 x 2 x 3 real line A random variable is random because before the random trial, we don’t know w hich basic outcome will occur, and therefore, we don t know which of the possible values of X will occur. X(e) e 2 e 1 e 6 A 2 e 3 e 7 e 4 e 5 A 1 A 3 S
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7 DISCRETE RANDOM VARIABLES The purpose of defining a random variable is to use it as a mathematical model for random quantities , such as number of customers sales (cars, mortgages, advertising spots) prices of stocks, bonds, and portfolios proportion of voters supporting a candidate number of patients cured by an experimental drug
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8 DISCRETE RANDOM VARIABLES By tracing the mapping, we can figure out all possible values of X (the set of possible realizations of X).
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