05c STAT discrete random variables 3-SHORT SLIDES

# 05c STAT discrete random variables 3-SHORT SLIDES -...

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1 DISCRETE RANDOM VARIABLES Dr. V.R. Bencivenga Economic Statistics Economics 329 DISCRETE RANDOM VARIABLES PART 3 BINOMIAL DISTRIBUTION

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2 DISCRETE RANDOM VARIABLES Introduction Quality control Does a shipment meet the standard for the maximum proportion of defective chips stated in the contract? Drug testing: Is an experimental drug better than older drugs? We’ll decide whether to accept the shipment based on the number of defectives in a sample. A key quantity for our decision is the probability we’d observe at least as many defectives as we find in our sample, if in fact the true proportion is what the contract says. We’ll conduct a drug trial, and report the results to the FDA (the number of patients who respond , and the number who don’t). A key quantity for the FDA’s decision is the probability we’d observe at least as many patients responding if in truth the experimental drug is no better than older drugs. I.e., what’s the probability of getting good results due to “luck?”
3 DISCRETE RANDOM VARIABLES The binomial random variable is a probability model for the number of “successes” in a “sequence” of success/failure trials. The DGP must have the following characteristics for the binomial random variable to be applicable: independent trials same “success probability” number of trials (n) decided in advance A probability model is a mathematical model of a random trial or “experiment.” In economics, a probability model is usually a random variable (or jointly-distributed random variables). The probability model consists of (i) the assumptions about the DGP necessary for the probability model to be relevant, and (ii) a set of events of interest, and the probabilities of those events. Here, conduct success/failure tries and count the number of successes. The binomial random variable is a probability model giving us the probabilities of different possible numbers of successes . A success/failure trial is called a Bernoulli trial.

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4 DISCRETE RANDOM VARIABLES Bernoulli trial and Bernoulli random variable A Bernoulli trial is a random trial with two basic outcomes (“success” and “failure”). To obtain a Bernoulli random variable , one basic outcome is assigned the value 1, and the other is assigned the value 0. 0 1 real line Test a drug on a patient, observe response/non-response: response non 0 response 1 X 1 Inspect an item from a production line, observe defective/non-defective: defective non 0 defective 1 X 1 X 1 e 2 e 1 S
5 DISCRETE RANDOM VARIABLES Derivation of the binomial probability function Some experiments can be described as a “sequence” of Bernoulli trials. Test a drug on a sequence of patients. Observe response/non-response of each. Inspect a sequence of items from a production line. Observe defective/non-defective for each.

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## This note was uploaded on 02/26/2012 for the course ECONOMICS 329 taught by Professor Bencivenga during the Spring '12 term at University of Texas.

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05c STAT discrete random variables 3-SHORT SLIDES -...

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