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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
Multiple Random Variables • Linear Combinations of Random Variables • Expected Value • Variance • Stochastic Models • Covariance of two Random Variables • Independence • Correlation
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
Population SPH&HS, UMASS Amherst 4
Population of N=3 Note: Population mean Variance.
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
Use as an Estimator: Sample Mean A Linear Estimator- a sum of random variables When n=2,
ID (s) Subject Response 1 Daisy 2 Lily 3 (=N) Rose Models for Response Non-Stochastic model (Deterministic) Stochastic model
Finite Population Pick a SRS with replacement of size n=2 Stochastic model SPH&HS, UMASS Amherst 10
Finite Population with replacement Stochastic model SPH&HS, UMASS Amherst 11
Finite Population with replacement Stochastic model SPH&HS, UMASS Amherst 12
Sampling- n=2 with replacement Stochastic model Random Variables Linear Combination of Random Variables SPH&HS, UMASS Amherst 13
Sampling- n=2 with replacement Realized Values SPH&HS, UMASS Amherst 14
Other Possible Samples with replacement SPH&HS, UMASS Amherst 15
Other Possible Samples with replacement SPH&HS, UMASS Amherst 16
Variance Matrix • When n=2, and SRS with replacement: Identity Matrix
Covariance of Random Variables When SRS without Replacment (n=2)
Covariance of two random variables when sampling without replacement
Estimating the Covariance Estimate the variance: • assuming srs Estimate the covariance: • assuming srs
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)
Example: SRS with rep n=2 Are and independent? Does ?
Sampling n=2 (with rep) Are and independent? Yes SPH&HS, UMASS Amherst 30
Sampling n=2 (with rep) Are and independent? Yes SPH&HS, UMASS Amherst 31
Sampling n=2 (with rep) Are and independent? Yes SPH&HS, UMASS Amherst 32
Example: SRS without rep n=2 Are and independent? Does ?
Sampling n=2 (without replacement) Are and independent? No SPH&HS, UMASS Amherst 34
Sampling n=2 (without replacement) Are and independent? No SPH&HS, UMASS Amherst 35
Sampling n=2 (without replacement) Are and independent? No SPH&HS, UMASS Amherst 36
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
Expected Value of a Linear Combination of Random Variables • Write linear combinations using vector notation. Random variables Constants
Example: SRS of size n: where
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.
Variance of a Linear Combination of Random Variables Example: Sample mean, n=2 srs with replacement Constants Random variables
Matrix Multiplication Hence
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
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