Содержание
- 2. Method of Distribution Functions X1,…,Xn ~ f(x1,…,xn) U=g(X1,…,Xn) – Want to obtain fU(u) Find values in
- 3. Example – Uniform X Stores located on a linear city with density f(x)=0.05 -10 ≤ x
- 4. Example – Sum of Exponentials X1, X2 independent Exponential(θ) f(xi)=θ-1e-xi/θ xi>0, θ>0, i=1,2 f(x1,x2)= θ-2e-(x1+x2)/θ x1,x2>0
- 5. Method of Transformations X~fX(x) U=h(X) is either increasing or decreasing in X fU(u) = fX(x)|dx/du| where
- 6. Example fX(x) = 2x 0≤ x ≤ 1, 0 otherwise U=10+500X (increasing in x) x=(u-10)/500 fX(x)
- 7. Method of Conditioning U=h(X1,X2) Find f(u|x2) by transformations (Fixing X2=x2) Obtain the joint density of U,
- 8. Example (Problem 6.11) X1~Beta(α=2,β=2) X2~Beta(α=3,β=1) Independent U=X1X2 Fix X2=x2 and get f(u|x2)
- 10. Method of Moment-Generating Functions X,Y are two random variables CDF’s: FX(x) and FY(y) MGF’s: MX(t) and
- 11. Sum of Independent Gammas
- 12. Linear Function of Independent Normals
- 13. Distribution of Z2 (Z~N(0,1))
- 14. Distributions of and S2 (Normal data)
- 15. Independence of and S2 (Normal Data) Independence of T=X1+X2 and D=X2-X1 for Case of n=2
- 16. Independence of and S2 (Normal Data) P2 Independence of T=X1+X2 and D=X2-X1 for Case of n=2
- 17. Distribution of S2 (P.1)
- 18. Distribution of S2 P.2
- 19. Summary of Results X1,…Xn ≡ random sample from N(μ, σ2) population In practice, we observe the
- 20. Order Statistics X1,X2,...,Xn ≡ Independent Continuous RV’s F(x)=P(X≤x) ≡ Cumulative Distribution Function f(x)=dF(x)/dx ≡ Probability Density
- 21. Order Statistics
- 22. Example X1,...,X5 ~ iid U(0,1) (iid=independent and identically distributed)
- 24. Distributions of Order Statistics Consider case with n=4 X(1) ≤x can be one of the following
- 25. Case with n=4
- 26. General Case (Sample of size n)
- 27. Example – n=5 – Uniform(0,1)
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