Functions of Random Variables

1. Functions of Random Variables

2. Method of Distribution Functions • X1,…,Xn ~ f(x1,…,xn) • U=g(X1,…,Xn) – Want to obtain fU(u) • Find values in (x1,…,xn) space where U=u • Find region where U≤u • Obtain FU(u)=P(U≤u) by integrating f(x1,…,xn) over the region where U≤u • fU(u) = dFU(u)/du

3. Example – Uniform X • Stores located on a linear city with density f(x)=0.05 -10 ≤ x ≤ 10, 0 otherwise • Courier incurs a cost of U=16X2 when she delivers to a store located at X (her office is located at 0)

4. Example – Sum of Exponentials • X1, X2 independent Exponential(q) • f(xi)=q-1e-xi/q xi>0, q>0, i=1,2 • f(x1,x2)= q-2e-(x1+x2)/q x1,x2>0 • U=X1+X2

5. Method of Transformations • X~fX(x) • U=h(X) is either increasing or decreasing in X • fU(u) = fX(x)|dx/du| where x=h-1(u) • Can be extended to functions of more than one random variable: • U1=h1(X1,X2), U2=h2(X1,X2), X1=h1-1(U1,U2), X2=h2-1(U1,U2)

6. Example • fX(x) = 2x 0≤ x ≤ 1, 0 otherwise • U=10+500X (increasing in x) • x=(u-10)/500 • fX(x) = 2x = 2(u-10)/500 = (u-10)/250 • dx/du = d((u-10)/500)/du = 1/500 • fU(u) = [(u-10)/250]|1/500| = (u-10)/125000 10 ≤ u ≤ 510, 0 otherwise

7. Method of Conditioning • U=h(X1,X2) • Find f(u|x2) by transformations (Fixing X2=x2) • Obtain the joint density of U, X2: • f(u,x2) = f(u|x2)f(x2) • Obtain the marginal distribution of U by integrating joint density over X2

8. Example (Problem 6.11) • X1~Beta(a=2,b=2) X2~Beta(a=3,b=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 MY(t) exist and equal for |t|<h,h>0 • Then the CDF’s FX(x) and FY(y) are equal • Three Properties: • Y=aX+b  MY(t)=E(etY)=E(et(aX+b))=ebtE(e(at)X)=ebtMX(at) • X,Y independent  MX+Y(t)=MX(t)MY(t) • MX1,X2(t1,t2) = E[et1X1+t2X2] =MX1(t1)MX2(t2) if X1,X2 are indep.

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 Thus T=X1+X2 and D=X2-X1 are independent Normals and & S2 are independent

17. Distribution of S2 (P.1)

18. Distribution of S2 P.2

19. Summary of Results • X1,…Xn≡ random sample from N(m, s2)population • In practice, we observe the sample mean and sample variance (not the population values: m, s2) • We use the sample values (and their distributions) to make inferences about the population values

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 Function • Order Statistics: X(1) ≤ X(2) ≤ ...≤ X(n) (Continuous  can ignore equalities) • X(1) = min(X1,...,Xn) • X(n) = max(X1,...,Xn)

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 cases: • Exactly one less than x • Exactly two are less than x • Exactly three are less than x • All four are less than x • X(3) ≤x can be one of the following cases: • Exactly three are less than x • All four are less than x • Modeled as Binomial, n trials, p=F(x)

25. Case with n=4

26. General Case (Sample of size n)

27. Example – n=5 – Uniform(0,1)