1 / 16

Multivariate Probability Distributions

Multivariate Probability Distributions. Multivariate Random Variables. In many settings, we are interested in 2 or more characteristics observed in experiments Often used to study the relationship among characteristics and the prediction of one based on the other(s)

rad
Download Presentation

Multivariate Probability Distributions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Multivariate Probability Distributions

  2. Multivariate Random Variables • In many settings, we are interested in 2 or more characteristics observed in experiments • Often used to study the relationship among characteristics and the prediction of one based on the other(s) • Three types of distributions: • Joint: Distribution of outcomes across all combinations of variables levels • Marginal: Distribution of outcomes for a single variable • Conditional: Distribution of outcomes for a single variable, given the level(s) of the other variable(s)

  3. Joint Distribution

  4. Marginal Distributions

  5. Conditional Distributions • Describes the behavior of one variable, given level(s) of other variable(s)

  6. Expectations

  7. Expectations of Linear Functions

  8. Variances of Linear Functions

  9. Covariance of Two Linear Functions

  10. Multinomial Distribution • Extension of Binomial Distribution to experiments where each trial can end in exactly one of k categories • n independent trials • Probability a trial results in category i is pi • Yi is the number of trials resulting in category I • p1+…+pk = 1 • Y1+…+Yk = n

  11. Multinomial Distribution

  12. Multinomial Distribution

  13. Conditional Expectations When E[Y1|y2] is a function of y2, function is called the regression of Y1 on Y2

  14. Unconditional and Conditional Mean

  15. Unconditional and Conditional Variance

  16. Compounding • Some situations in theory and in practice have a model where a parameter is a random variable • Defect Rate (P) varies from day to day, and we count the number of sampled defectives each day (Y) • Pi ~Beta(a,b) Yi |Pi ~Bin(n,Pi) • Numbers of customers arriving at store (A) varies from day to day, and we may measure the total sales (Y) each day • Ai ~ Poisson(l) Yi|Ai ~ Bin(Ai,p)

More Related