1 / 62

45-733: lecture 6 (chapter 5)

45-733: lecture 6 (chapter 5). Continuous Random Variables. Joint continuous distributions. The joint continuous distribution is a complete probabilistic description of a group of r.v.s Describes each r.v. Describes the relationship among r.v.s. Joint continuous distributions.

abe
Download Presentation

45-733: lecture 6 (chapter 5)

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. 45-733: lecture 6 (chapter 5) Continuous Random Variables William B. Vogt, Carnegie Mellon, 45-733

  2. Joint continuous distributions • The joint continuous distribution is a complete probabilistic description of a group of r.v.s • Describes each r.v. • Describes the relationship among r.v.s William B. Vogt, Carnegie Mellon, 45-733

  3. Joint continuous distributions • Cumulative distribution function • The cdf of a group of r.v.s describes the probability that each of them will be less than specified values, simultaneously: William B. Vogt, Carnegie Mellon, 45-733

  4. Joint continuous distributions • Cumulative distribution function • Graphing joint cdfs is difficult, because such a graph requires at least 3 dimensions • I will instead try to communicate joint distribution information through the use of scatterplots William B. Vogt, Carnegie Mellon, 45-733

  5. Joint continuous distributions • Cumulative distribution function • Scatterplots • No relationship X and Y Y X William B. Vogt, Carnegie Mellon, 45-733

  6. Joint continuous distributions • Cumulative distribution function • Scatterplots • A relationship between X and Y Y X William B. Vogt, Carnegie Mellon, 45-733

  7. Joint continuous distributions • Cumulative distribution function • Scatterplots • A relationship between X and Y Y X William B. Vogt, Carnegie Mellon, 45-733

  8. Joint continuous distributions • Cumulative distribution function • Scatterplots • A relationship between X and Y Y X William B. Vogt, Carnegie Mellon, 45-733

  9. Joint continuous distributions • Marginal distribution function • The marginal distribution function of each of a group of r.v.s is just its cdf: William B. Vogt, Carnegie Mellon, 45-733

  10. Joint continuous distributions • Independence • A group of continuous random variables is independent if their joint cdf factors into the individual cdfs: William B. Vogt, Carnegie Mellon, 45-733

  11. Joint continuous distributions • Covariance • Covariance measures the (linear) association between two random variables, just as in the discrete case. • Covariance is the expectation of the product of each variable’s deviation from its mean: William B. Vogt, Carnegie Mellon, 45-733

  12. Joint continuous distributions • Covariance • Covariance is positive when the two variables tend to be above their means together and below their means together • Covariance is negative when one variable tends to be above its mean when the other is below. William B. Vogt, Carnegie Mellon, 45-733

  13. Joint continuous distributions • Covariance • A positive covariance between X,Y Y X William B. Vogt, Carnegie Mellon, 45-733

  14. Joint continuous distributions • Covariance • A negative covariance between X,Y Y X William B. Vogt, Carnegie Mellon, 45-733

  15. Joint continuous distributions • Covariance • A zero covariance between X,Y • X,Y are independent Y X William B. Vogt, Carnegie Mellon, 45-733

  16. Joint continuous distributions • Covariance • A zero covariance between X,Y • X,Y are not independent Y X William B. Vogt, Carnegie Mellon, 45-733

  17. Joint continuous distributions • Covariance • A zero covariance between X,Y • X,Y are not independent Y X William B. Vogt, Carnegie Mellon, 45-733

  18. Joint continuous distributions • Covariance • A zero covariance between X,Y • X,Y are not independent Y X William B. Vogt, Carnegie Mellon, 45-733

  19. Joint continuous distributions • Covariance • As before, if X,Y are independent, then Cov(X,Y)=0 • As before, it is not the case that a zero covariance between X,Y implies that X and Y are independent William B. Vogt, Carnegie Mellon, 45-733

  20. Joint continuous distributions • Sums: • The rules for sums of continuous rvs and expectation, variance are the same as for discrete: William B. Vogt, Carnegie Mellon, 45-733

  21. Joint Distributions • Sums and differences • The expectation of sums is the sum of expectations William B. Vogt, Carnegie Mellon, 45-733

  22. Joint Distributions • Sums and differences • The variance of a sum is the sum of variances, plus twice the covariance William B. Vogt, Carnegie Mellon, 45-733

  23. Joint Distributions • Sums and differences • If two variables are uncorrelated (covariance is 0), the variance of a sum is the sum of variances William B. Vogt, Carnegie Mellon, 45-733

  24. Joint Distributions • Sums and differences • The variance of a sum is the sum of variances plus twice the sum of every possible covariance: William B. Vogt, Carnegie Mellon, 45-733

  25. Joint Distributions • Sums and differences • The variance of a sum is the sum of variances plus twice the sum of every possible covariance: William B. Vogt, Carnegie Mellon, 45-733

  26. Joint Distributions • Sums and differences • The covariance of a sum with a third variable is the sum of covariances: William B. Vogt, Carnegie Mellon, 45-733

  27. Normal distribution • The normal distribution is the most important distribution in statistics • Many interesting and useful random variables follow the normal distribution • Many interesting and useful random variables do not follow the normal distribution but may be approximated by it William B. Vogt, Carnegie Mellon, 45-733

  28. An aside: parameters • Recall that the distribution of a random variable contains all the information which statistics can discover about it • Distribution may be expressed in a probability function or a cumulative distribution function for a discrete random variable • Distribution may be expressed in a density function or a cumulative distribution function for a continuous random variable William B. Vogt, Carnegie Mellon, 45-733

  29. An aside: parameters • Often, the information in a distribution is too much to process easily • When this is true, we want summary measures of the information: • Mean • Variance • Median • Percentiles William B. Vogt, Carnegie Mellon, 45-733

  30. An aside: parameters • These summary measures are often called parameters of the distribution • Mean is often written  when we are thinking of it as a parameter • Standard deviation is often written  when we are thinking of it as a parameter William B. Vogt, Carnegie Mellon, 45-733

  31. An aside: parameters • Also, recall that probability functions, density functions, cdfs often depend on unspecified numbers • These numbers are also called parameters William B. Vogt, Carnegie Mellon, 45-733

  32. An aside: parameters William B. Vogt, Carnegie Mellon, 45-733

  33. Normal distribution • Density function • Two parameters,  and  • The formula for the density function: William B. Vogt, Carnegie Mellon, 45-733

  34. Normal distribution • Density function • Two parameters,  and  • It turns out that if X is distributed normal: William B. Vogt, Carnegie Mellon, 45-733

  35. Normal distribution • Density function for =0, =1 William B. Vogt, Carnegie Mellon, 45-733

  36. Normal distribution • Density functions for =1 =2 =5 William B. Vogt, Carnegie Mellon, 45-733

  37. Normal distribution • Density functions for =1 =2 =1 William B. Vogt, Carnegie Mellon, 45-733

  38. Normal distribution • Standard normal • The normal distribution with mean zero and variance one has a special name, the standard normal William B. Vogt, Carnegie Mellon, 45-733

  39. Normal distribution • Cumulative distribution function • There is no nice formula for it • The standard normal cdf is compiled in tables • We write the standard normal cdf: William B. Vogt, Carnegie Mellon, 45-733

  40. Normal distribution • The normal distribution is symmetric -a 0 a William B. Vogt, Carnegie Mellon, 45-733

  41. Normal distribution • The normal table • Contains tabulations of the standard normal cdf. • Different tables tabulate the cdf differently William B. Vogt, Carnegie Mellon, 45-733

  42. Normal distribution • The normal table • Our book’s normal table provides literally the cdf, (z)=P(Zz): 0 z William B. Vogt, Carnegie Mellon, 45-733

  43. Normal distribution • The normal table • Another common table type: 0 z William B. Vogt, Carnegie Mellon, 45-733

  44. Normal distribution • Using the normal table • If you want to know the probability that Z, distributed N(0,1) is less than some value, just look up the value in the table • P(Z1)=0.8413 • P(Z2.03)=0.9788 • P(Z-2)=?? William B. Vogt, Carnegie Mellon, 45-733

  45. Normal distribution • Using the normal table • Our table has no negative numbers! • P(Z-2)=?? • P(Z-2)=P(Z>2)=1-P(Z -2) • P(Z-2)=1-0.9772 • P(Z-2)=0.0228 -2 0 2 William B. Vogt, Carnegie Mellon, 45-733

  46. Normal distribution • Using the normal table • How about: • P(Z>2)=1-P(Z -2) • P(Z>2)=1-0.9772 • P(Z>2)=0.0228 -2 0 2 William B. Vogt, Carnegie Mellon, 45-733

  47. Normal distribution • Using the normal table • Ranges • P(1Z2)= P(Z2)- P(Z1) • P(1Z2)= 0.9772-0.8413 • P(1Z2)= 0.1359 0 2 William B. Vogt, Carnegie Mellon, 45-733

  48. Normal distribution • Using the normal table • What if X is distributed N(1,4) and you need to know P(X<5)? • Can only use the table for random variables distributed N(0,1) William B. Vogt, Carnegie Mellon, 45-733

  49. Normal distribution • Using the normal table • Must “standardize” X: William B. Vogt, Carnegie Mellon, 45-733

  50. Central limit theorem • The central limit theorem says (in essence) that the standardized average of a bunch of random variables tends to be distributed normal as the number of random variables grows large. William B. Vogt, Carnegie Mellon, 45-733

More Related