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Functions of Random Variables 2. Method of Distribution Functions презентация
Содержание
- 1. Functions of Random Variables 2. Method of Distribution Functions
- 2. Method of Distribution Functions X1,…,Xn ~ f(x1,…,xn)
- 3. Example – Uniform X Stores located on
- 4. Example – Sum of Exponentials X1, X2
- 5. Method of Transformations X~fX(x) U=h(X) is either
- 6. Example fX(x) = 2x 0≤ x ≤
- 7. Method of Conditioning U=h(X1,X2) Find f(u|x2) by
- 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
- 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
- 16. Independence of and S2
- 17. Distribution of S2 (P.1)
- 18. Distribution of S2 P.2
- 19. Summary of Results X1,…Xn ≡ random sample
- 20. Order Statistics X1,X2,...,Xn ≡ Independent Continuous RV’s
- 21. Order Statistics
- 22. Example X1,...,X5 ~ iid U(0,1) (iid=independent and identically distributed)
- 24. Distributions of Order Statistics Consider case with
- 25. Case with n=4
- 26. General Case (Sample of size n)
- 27. Example – n=5 – Uniform(0,1)
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
Слайд 2Method 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
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
Слайд 3Example – 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)
Courier incurs a cost of U=16X2 when she delivers to a store located at X (her office is located at 0)
Слайд 4Example – 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
U=X1+X2
f(x1,x2)= θ-2e-(x1+x2)/θ x1,x2>0
U=X1+X2
Слайд 5Method 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)
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)
Слайд 6Example
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
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
Слайд 7Method 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
f(u,x2) = f(u|x2)f(x2)
Obtain the marginal distribution of U by integrating joint density over X2
Слайд 8Example (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)
Слайд 10Method 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|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.
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.
Слайд 16Independence 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
Слайд 19Summary of Results
X1,…Xn ≡ random sample from N(μ, σ2) population
In practice,
we observe the sample mean and sample variance (not the population values: μ, σ2)
We use the sample values (and their distributions) to make inferences about the population values
We use the sample values (and their distributions) to make inferences about the population values
Слайд 20Order 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)
Order Statistics: X(1) ≤ X(2) ≤ ...≤ X(n) (Continuous ⇒ can ignore equalities)
X(1) = min(X1,...,Xn)
X(n) = max(X1,...,Xn)
Слайд 24Distributions 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)
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)
