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Basic Probability & Random Variables

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Axioms of Probability

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Conditional Probability

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Multiplicative Rule of Probability

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Bayes Rule is very important

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Here is a useful application of Bayes

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Randomization Response Theory Assume that you need estimate the proportion of narcotic drug consumption among university students. It is unlikely that students would answer your questionnaire honestly. So here is a simple trick you may use. Instead of asking the question directly, let the student draw a ball from an urn in which there are 8 blue and 2 yellow balls. If a yellow ball is drawn (you do not see the result), student answers the question «Is the last digit of your TC ID number odd?» and if a blue ball is drawn then the student answers «Have you ever used a narcotic drug?» question.

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Soon we will be able to compute the error due to …(?)

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Random Variables and Probability Distributions Random Variable:? Example: When rolling a two dice, we may be interested in whether or not the sum of the two dice is 7. Or we might be interested in the sum of the two dice.

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Probability Density Function In class: How probabilities are related with areas under the curve.

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Expectation

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Variance

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Typically you need to know what sort of probability distributions are there and for which type of situations thay are used for. We will be mostly dealing with Normal Distribution.

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INCOME DISTRIBUTION – (Empirical)

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INCOME DISTRIBUTION – (Theoretical) Log Normal Didtribution

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Normal Distribution x z Standard Normal Distribution

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P(x < 500) = P(z < 1) Normal Distribution 600μ =500 P(x < 600) μ = 500 σ = 100 x Standard Normal Distribution 1μ = 0 μ = 0 σ = 1 z P(z < 1) Same Area

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Sampling Probability Sampling Nonprobability Sampling

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Probability Sampling Sampling element Population Target population Sampling frame Sampling ratio

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There is a classic Jimmy Stewart movie, Magic Town, about "Grandview," a small town in the Midwest that is a perfect statistical microcosm of the United States, a place where the citizens' opinions match perfectly with Gallup polls of the entire nation. A pollster (Jimmy Stewart), secretly uses surveys from this "mathematical miracle" as a shortcut to predicting public opinion. Instead of collecting a national sample, he can more quickly and cheaply collect surveys from this single small town. The character played by Jane Wyman, a newspaper editor, finds out what is going on and publishes her discovery. As a result the national media descend upon the town, which becomes, overnight, "the public opinion capital of the U.S."

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Probability Sampling

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Check POPULATION PARAMETERSSAMPLE STATISTICS To be filled in class

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Sampling Distribution

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Probability Sampling Random sample Sampling error Four Ways to Sample Randomly – Simple Random – Systematic – Stratified Sampling – Cluster Sampling

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Random Sample Sampling Error: Variation Component Sample size Component

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Sampling Distribution and Sampling Error

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Sampling and Confidence x

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