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Econometrics

Econometrics. Chapter 3 Random Variables. Random Variables– Chapter 3. Random Variable Takes a single, specific value That value is unknown until it happens All possible values ARE known The probability that the RV takes one of these possible values is known (Dice example) Describing RVs

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Econometrics

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  1. Econometrics Chapter 3 Random Variables

  2. Random Variables– Chapter 3 • Random Variable • Takes a single, specific value • That value is unknown until it happens • All possible values ARE known • The probability that the RV takes one of these possible values is known • (Dice example) • Describing RVs • RV can be completely described (listing all possible values or listing a range of values it can take along with the chance that it will take any one of them) • The realized value of the RV is the observed value

  3. Random Variables – continuedProbability Density Function • Discrete Random Variables • A random variable that takes a finite number of values • Chart showing all possible values and their probabilities is a probability mass function (PMF) • Continuous Random Variables • A random variable that may take any value within a range of values • Graph showing the probability that the RV takes a value within that range is the Probability Density Function (PDF)

  4. Random Variables – continuedProbability Density Function • PDFs (cont) • Probability that any specific value of a continuous RV will occur = 0 (since it may take an infinite number of values between any 2 other values) • Interested in the chance that the value of a continuous RV will fall within a range of values • PDF is defined so that the area under its curve is the probability the RV will fall in that range

  5. Random Variables – continuedCumulative Distribution Function • Sometimes interested not with the chance that a RV will take a specific value, but that it will be less than (or equal to) a certain value • Chance of Rolling 4 or Less is 1/6 x 4 = 2/3 = 66.7% • CDF of RV “X” is a graph associating all possible values (or ranges of values) with the chance that “X” will be less than or equal to “X”

  6. Random Variables – continuedCumulative Distribution Function

  7. Random Variables – continuedProperties of Random Variables • PDF completely describes RV • Not very useful mathematically • Expected value or mean value • Summarizes information about a RV from PDF • Value we expect to find for RV (on average) • For a Continuous RV: • For a Discrete RV:

  8. Random Variables – continuedProperties of Random Variables • Theorem – Law Of Large Numbers • If we repeatedly observe realized (actual) values of random variables and calculate the average of those values, this average will tend to be close to the RV’s expected value. And the more times we observe the random variable, the closer to the average the expected value will be.

  9. Random Variables – continuedProperties of Random Variables • Variance • The second (after mean) measure of a RV • Measures how far from its mean a RV is likely to be • It’s the average value of the squared distance between the observed value of the RV and its mean • Standard Deviation • The square root of the variance

  10. Random Variables – continuedNormal Density Function • Probability Density Functions (PDF) – Normal Distribution or Bell Curve • One of the most commonly used PDFs in economics • 50% probability of finding a value of a RV less than or equal to the RVs mean (μ)

  11. Random Variables – continuedNormal Density Function • Standard Normal Distribution • Same as “Bell Curve” except this PDF has a Mean = 0 and a Variance = 1 • When μ=0 and σ2=1,

  12. Random Variables – continuedNormal Density Function • General PDF for a Normally Distributed RV with Mean = μ and Variance = σ2

  13. Random Variables – continuedNormal Density Function • Language/Description of a RV • If the RV “X” is distributed normally with a mean of μ and a variance σ2 • X  N ( μ,σ2 )

  14. Random Variables – continuedJoint Distribution of Two Random Variables • If two RVs are related→ • The value that one takes depends, at least in part, on the value the other takes • Knowing the value that one RV takes helps predict the value the other one will take • Covariance: Positive (move together) or Negative (opposite) or 0 (independent) • Correlation: covariance divided by standard deviations

  15. Random Variables - continuedSummary – Chapter 3 • Random Variables: Actual values are unknown until measured, but all possible values and the probability of those values ARE known • Realized Value: The value the RV actually takes when measured • Probability Density Function (PDF): Graph of all possible values and the chance that value will occur

  16. Random Variables - continuedSummary – Chapter 3 • Cumulative Distribution Function (CDF): Graph of all possible values of a RV and the chance that the realized value will be less than that value • Mean Value: A measure of the value a RV is expected to take • Standard Deviation: A measure of how close a RV will be to its mean • Variance: The squared standard deviation

  17. Random Variables - continuedSummary – Chapter 3 • Normal Random Variables: RV that have a PDF of the “Bell Curve” • Covariance and Correlation: Measures the extent to which the realized value of two RVs tend to rise and fall together or oppositely

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