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Mastering Continuous Random Variables for Statistical Analysis

Explore the intricacies of handling real-valued random variables and the distinction between probability mass and density functions in statistical analysis. Learn practical tips for managing continuous random variables effectively.

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Mastering Continuous Random Variables for Statistical Analysis

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  1. Continuous Random Variables (most slides borrowed with permission from Andrew Moore of CMU and Google) http://www.cs.cmu.edu/~awm/tutorials

  2. Announcements • CS Welcome event • Thursday 3:30, ECCR 265 • poster presentations • Mozer lab research meeting • Wednesdays 11:00-12:30, ECCS 127 • Email me if you’d like to be on our mailing list

  3. Real-Valued Random Variables • Previous lecture on probability focused on discrete random variables • true, false • male, female • freshman, sophomore, junior, senior • Can sometimes quantize real variables to make them discrete • E.g., age, height, distance • Today: how to handle variables that cannot be quantized

  4. Probability Mass Vs. Density • Discreet RVs have a probability mass associated with each value of the variable • P(male)=.7, P(female)=.3 • Imagine if the variablehad an infinitenumber of valuesinstead of a finitenumber…

  5. Probability Mass Vs. Density • Continuous RVs have a probability density associated with each value • Probability density function (PDF) • Density is derivative of mass • Notation: P(…) for mass,p(…) for density

  6. = E[X2] - E[X]2

  7. Density estimate of automobile weight and MPG Note change innotation: Previously used P(x^y) for joint

  8. Covariance Facts Consider 2D case with (X,Y) FALSE TRUE ? ?

  9. Mike’s Basic Advice on Continuous Random Variables • Ignore the fact that p(x) is a probability density function and treat it just as a mass function, and the algebra all works out. • Alternatively, turn densities to masses with dx terms, and they should always cancel out. • Don’t be freaked when you see a probability density >> 1. • Do be freaked if you see a probability mass or density < 0.

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