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Unformatted text preview: uppose they have this relationship: E(GPA|StudTime) = 1+ 0.1StudTime. Further, we know that E(StudTime) = 15 hours. Then E(GPA) = E(E(GPA|StudTime))=E(1 + 0.1StudTime) = 1 + 0.1(15) = 2.5 2. E(Wage|Educ) = 3.5 + .7Educ, and E(Educ) = 12 Then E(Wage) = 11.9 Conditional Variance The variance of Y given that you know X=x. Var (Y | X ) = E (Y 2 | X ) − [ E (Y | X )]2 Facts about conditional variance 1. If X and Y are independent, then Var(Y|X)=Var(Y) Example: if Y=1+0.1X+u, where X and u are independent, then Var(Y|X)=Var(u). Sample mean, sample variance, sample covariance, sample correlation Recall: A sample is a group of observations (observed outcomes, realizations on one or more random variables). Can always draw another sample from the population, so the sample mean, sample variance, sample covariance etc all depend on the specific sample you have. Given any samples of size n for a r.v. X, ˆ
• Sample mean: μ X = X = 1n
∑ xi ,
n i =1 where 1/n is the prob, as we see each realization once in the sample. ˆX
• Sample variance: σ 2 = 1n
∑ ( xi − X )2 n − 1 i =1 ˆ
• Sample sd: σ X = σ X 2 ˆ
• Sample covariance: σ XY = 1n
∑ ( xi − X )( yi − Y ) n − 1 i =1 8 ˆ
• Sample correlation: ρ XY = ˆ
σ XY ˆˆ
σ XσY Estimator, Estimate, and Sampling Distribution Estimators and Estimates An estimator is a rule (typically a formula) applied to a sample of data to estimate some unknown population parameter. Example: Use sample mean, sample variance, and Sample covariance to estimate unknown population parameters, population mean, population variance, and covariance. Sample mean is an estimator of the population mean. Sample variance (sd) is an estimator of the population variance (sd). Sample covariance is an estimator of the population covariance. An estimator therefore can take on different values, depending on the sample, so •
• An estimator is a r.v., An estimator has a distribution. Immediately we have, sample mean, variance etc are all r.v.s The specific value you obtain is just one observation on the r.v. Thought experiments 1. Randomly pick 100 students fro...
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This note was uploaded on 02/17/2014 for the course PSY 2000 taught by Professor Staff during the Spring '08 term at University of Texas at Austin.
- Spring '08