Stat 410: Random Variables and Confidence Intervals - Review of Dr. D. Scott's Lecture, Study notes of Statistics

Download Stat 410: Random Variables and Confidence Intervals - Review of Dr. D. Scott's Lecture and more Study notes Statistics in PDF only on Docsity! Stat 410 Random Variable and Confidence Interval Review Dr. D. Scott August 23, 2005 Fundamental object is the density function: X ∼ f(x) = f(x1, x2, . . . , xp), which encodes ”structure.” Sometimes one of the variables is labeled dif- ferently: (X, Y ) ∼ f(x, y) = f(x1, . . . , xp, y). Here, Y is the dependent or response vari- able, while X1, . . . , Xp are the independent or predictor variables. σ2x̄ = E(X̄ − µx̄)2 = E   1 n n ∑ i=1 (Xi − µ)   2 = 1 n2 E   n ∑ i=1 (Xi − µ)2 + ∑ i 6=j (Xi − µ)(Xj − µ)   = 1 n2 · n σ2x + 1 n2 n(n − 1) · 0 · 0 Why? = σ2x n . Normal Results Z ∼ N(0,1) X ∼ N(µ, σ2) f(x) = 1√ 2π σ exp ( − (x − µ) 2 2σ2 ) Facts: X − µ σ ∼ N(0,1) (exactly) X̄ ∼ N(µ, σ 2 n ) X̄ − µ σ/ √ n ∼ N(0,1) Z2 ∼ χ2(1) where Z ∼ N(0,1) n ∑ i=1 Z2i ∼ χ2(n) . CLT: ∑ of i.i.d. r.v.’s ≈ Normal. (RVLS) Confidence Intervals for Parameters: pivots Rearrange Prob(−1.96 < X̄ − µ σ/ √ n < 1.96) = 95% to get Prob(X̄ −1.96 σ√ n < µ < X̄ +1.96 σ√ n ) = 95% (±2.576 for a 99% confidence interval) Since E[χ2(p)] = p and V ar[χ2(p)] = 2p, E [ (n − 1) σ2 S2 ] = n − 1 or E[S2] = σ2 (unbiased) Now have a pivot for a C.I. for σ2: Prob ( a < (n − 1)S2 σ2 < b ) = 1 − α iff Prob ( (n − 1)S2 b < σ2 < (n − 1)S2 a ) , where Pr(χ2n−1 < a) = Pr(χ 2 n−1 > b) = α 2 . (note: show R code to plot χ2(6)) Need a better pivot for µ, since do not really know σ2 for the normal random variable. Obvious idea is to use S2 in place of σ2! S2 = 1 n − 1 ∑ (Xi − X̄)2 S2(X̄) = 1 n S2 = 1 n(n − 1) ∑ (Xi − X̄)2 is unbiased for σ2x̄. Relationship to hypothesis testing. H0 : µ = µ0 vs. H1 : µ 6= µ0 Compute T̂ = (X̄ − µ0)/S(X̄) and reject if |T̂ | > a. Same as checking if µ0 is in C.I. (symmetry)! p-value = Prob(|Tn−1| > |T̂ |) (learn) Finally, the F distribution is a pivot for com- paring variances. F = χ2(ν1)/ν1 χ2(ν2)/ν2 ∼ F(ν1, ν2) = Fν1,ν2 Recall: Tn−1 = Z √ χ2n−1/(n − 1) Thus T2n−1 = Z2 χ2n−1/(n − 1) = χ2(1) χ2n−1/(n − 1) = F1,n−1