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Introduction to Biostatistics (PUBHLTH 540) Multiple Random Variables

Introduction to Biostatistics (PUBHLTH 540) Multiple Random Variables. Multiple Random Variables. Linear Combinations of Random Variables Expected Value Variance Stochastic Models Covariance of two Random Variables Independence Correlation. An Example.

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Introduction to Biostatistics (PUBHLTH 540) Multiple Random Variables

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  1. Introduction to Biostatistics (PUBHLTH 540)Multiple Random Variables

  2. Multiple Random Variables • Linear Combinations of Random Variables • Expected Value • Variance • Stochastic Models • Covariance of two Random Variables • Independence • Correlation

  3. An Example Choose a Simple Random Sample with Replacement of size n=2 from a Population of N=3 Observe: 1 Response (i.e. Age) on each Subject in the Sample Question: What is the average age of subjects in the population? Use the sample mean to estimate the Population Average Age Introducing…. Daisy Lily Rose SPH&HS, UMASS Amherst 3

  4. Population SPH&HS, UMASS Amherst 4

  5. Population of N=3 Note: Population mean Variance.

  6. Pick SRS with Replacement of n=2 a random variable representing the 1st selection i=1,…,n=2 a random variable representing the 2nd selection

  7. Use as an Estimator: Sample Mean A Linear Estimator- a sum of random variables When n=2,

  8. Linear Combination of Random Variables Example: Sample Mean

  9. ID (s) Subject Response 1 Daisy 2 Lily 3 (=N) Rose Models for Response Non-Stochastic model (Deterministic) Stochastic model

  10. Finite Population Pick a SRS with replacement of size n=2 Stochastic model SPH&HS, UMASS Amherst 10

  11. Finite Population with replacement Stochastic model SPH&HS, UMASS Amherst 11

  12. Finite Population with replacement Stochastic model SPH&HS, UMASS Amherst 12

  13. Sampling- n=2 with replacement Stochastic model Random Variables Linear Combination of Random Variables SPH&HS, UMASS Amherst 13

  14. Sampling- n=2 with replacement Realized Values SPH&HS, UMASS Amherst 14

  15. Other Possible Samples with replacement SPH&HS, UMASS Amherst 15

  16. Other Possible Samples with replacement SPH&HS, UMASS Amherst 16

  17. All Possible Samples

  18. Expected Values

  19. Covariance of Two Random Variables

  20. Based on simple random sampling with replacement

  21. Variance Matrix • When n=2, and SRS with replacement: Identity Matrix

  22. Variance Matrix for n Random Variables

  23. Covariance of Random Variables When SRS without Replacment (n=2)

  24. Covariance of two random variables when sampling without replacement

  25. Estimating the Covariance Estimate the variance: • assuming srs Estimate the covariance: • assuming srs

  26. Independence • Two random variables, Y and Z are independent if P(Y=y|Z=z)=P(Y=y) P(Y=y|Z=z) means the probability that Y has a value of y, given Z has a value of z (see Text, sections 6.1 and 6.2)

  27. Example: SRS with rep n=2 Are and independent? Does ?

  28. Sampling n=2 (with rep) Are and independent? Yes SPH&HS, UMASS Amherst 30

  29. Sampling n=2 (with rep) Are and independent? Yes SPH&HS, UMASS Amherst 31

  30. Sampling n=2 (with rep) Are and independent? Yes SPH&HS, UMASS Amherst 32

  31. Example: SRS without rep n=2 Are and independent? Does ?

  32. Sampling n=2 (without replacement) Are and independent? No SPH&HS, UMASS Amherst 34

  33. Sampling n=2 (without replacement) Are and independent? No SPH&HS, UMASS Amherst 35

  34. Sampling n=2 (without replacement) Are and independent? No SPH&HS, UMASS Amherst 36

  35. Relationship between Independence and Covariance • If two random variables are independent, then their covariance is 0. • If the covariance of two random variables is zero, the two may (or may not) be independent

  36. Expected Value of a Linear Combination of Random Variables • Write linear combinations using vector notation. Random variables Constants

  37. Example: SRS of size n: where

  38. Example 2: Suppose two independent SRS w/o replacement are selected from populations of boy and girl babies, and the weight recorded. Let us represent the boy weight by Y and the girl weight by X. Suppose sample results are given as follows: An estimate is wanted of the average birth weight in Europe, where for every 1000 births, 485 are girls, while 515 are boys. Write a linear combination that can be used to construct an estimator.

  39. Variance of a Linear Combination of Random Variables Example: Sample mean, n=2 srs with replacement Constants Random variables

  40. Matrix Multiplication Hence

  41. Practice: Variance of a Linear Combination of Random Variables Example: Sample mean, n=2 srs withOUT replacement from a population of N Random variables Constants

  42. Correlation (see 17.1, 17.2 in text) • The correlation between two random variables is defined as • Based on a simple random sample, we estimate the correlation by

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