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Basic Probability & Random Variables. Axioms of Probability.

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Presentation on theme: "Basic Probability & Random Variables. Axioms of Probability."— Presentation transcript:

1 Basic Probability & Random Variables

2 Axioms of Probability

3 Conditional Probability

4 Multiplicative Rule of Probability

5 Bayes Rule is very important

6 Here is a useful application of Bayes

7 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.

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

9 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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12 Probability Density Function In class: How probabilities are related with areas under the curve.

13 Expectation

14

15 Variance

16 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.

17 INCOME DISTRIBUTION – (Empirical)

18 INCOME DISTRIBUTION – (Theoretical) Log Normal Didtribution

19

20 Normal Distribution x     z Standard Normal Distribution

21 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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23 Sampling Probability Sampling Nonprobability Sampling

24 Probability Sampling Sampling element Population Target population Sampling frame Sampling ratio

25 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."

26 Probability Sampling

27 Check POPULATION PARAMETERSSAMPLE STATISTICS To be filled in class

28 Sampling Distribution

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

30 Random Sample Sampling Error: Variation Component Sample size Component

31 Sampling Distribution and Sampling Error

32 Sampling and Confidence x


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