ecn10-lecture7-v3

ecn10-lecture7-v3 - Economics 10: Introduction to...

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Economics 10: Introduction to Statistical Methods Lecture #5 Introduction to Probability, Probability Models and Rules
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So Far Summarizing univariate and bivariate data Working with samples; want to use samples to learn about the population This process: statistical inference To understand statistical inference and the linear regression model, need to understand probability
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Today’s Lecture Probability models Set theory notation Probability rules
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Probability Models
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Random trials A random trial is an activity where several outcomes are possible and which outcome will prevail is not known a priori (before the trial) A random variable is a variable whose value is a numerical outcome of a random trial (or phenomenon) The activity can be many things, e.g.,: Flipping a coin or rolling a die Flipping a coin or rolling a die n times The answer to a survey question Survey responses from random sample of size n
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Coin toss The result of any single coin toss is random. But the result over many tosses is predictable, as long as the trials are independent (i.e., the outcome of a new coin flip is not influenced by the result of the previous flip). First series of tosses Second series The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.
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The trials are independent only when you put the coin back each time. It is called sampling with replacement. Two events are independent if the probability that one event occurs on any given trial of an experiment is not affected or changed by the occurrence of the other event. When are trials not independent? Imagine that these coins were spread out so that half were heads up and half were tails up. Close your eyes and pick one. The probability of it being heads is 0.5. However, if you don’t put it back in the pile, the probability of picking up another coin that is heads up is now less than 0.5.
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What is a probability model? A probability model consists of: A sample space – list of all possible outcomes of a random trial Probability with which each of these outcomes occurs The probability that an outcome occurs is the fraction of times it occurs in repeated random trials
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Sample space A sample space , S , of a random trial is the set of all possible outcomes of the trial Example 1: Rolling a die once S = {1, 2, 3, 4, 5, 6} Example 2: Flipping a coin two times and recording results in order S = {HH,HT,TH,TT}
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Sample Space Example 3: Response to survey question about labor force status: S = {employed, unemployed, not in labor force, no response} Example 4: Survey question about wage S=[0, )
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Outcomes and events An outcome is one element of the sample space An event is a set of outcomes of a random trial, or a subset of the sample space A particular outcome is one possible event Example: Let event A be “at least one head in two flips of a coin”. Then A = {HH,HT,TH}
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Set Theory Notation
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ecn10-lecture7-v3 - Economics 10: Introduction to...

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