Let X1, . . . , X4 be independent Gamma(1, 1) random variables. Let X(1), ....

Question

Question

Let X₁, . . . , X₄ be independent Gamma(1, 1) random variables. Let X₍₁₎, . . . , X₍₄₎ denote their order statistics.

(a) Show that Y₁ = X₍₁₎, Y₂ = X₍₂₎ − X₍₁₎, Y₃ = X₍₃₎ − X₍₂₎ and Y₄ = X₍₄₎ − X₍₃₎ are independent, with Y_j = Gamma(1, 1/(5 − j)), j = 1, 2, 3, 4.

(b) Use this to find the dispersion matrix of the order statistics.

math Statistics-And-Probability

Answer 1

Answer #1

Answer 2

Similar Homework Help Questions

Let Y1, Y2, and Y3 be independent, N(0, 1)-distributed random variables, and set X1 = Y1...

Let Y1, Y2, and Y3 be independent, N(0, 1)-distributed random variables, and set X1 = Y1 − Y3, X2 = 2Y1 + Y2 − 2Y3, X3 = −2Y1 + 3Y3.Determine the conditional distribution of X2 given that X1 + X3 = x.
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y;...

Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...
Let Y1 N(1,1), Y2 N(2,2), and Y3 N(3,3) be independent random variables. Find a new random...

Let Y1 N(1,1), Y2 N(2,2), and Y3 N(3,3) be independent random variables. Find a new random variable Y4 that is a function of Y1, Y2, and Y3 such that Y4 has a t-distribution with 2 degrees of freedom, and explain why it has that distribution. (To avoid confusion, the parameters in the normal distributions above are the mean and the standard deviation.)

Let X1 and X2 be independent gamma distribution random variables with gamma (a1,1) and gamma (a2,...

Let X1 and X2 be independent gamma distribution random variables with gamma (a1,1) and gamma (a2, 1). Find the marginal distributions of x1/(x1+x2) and x2/(x1+x2).
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) =...

3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... <...

1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... < Yn be the order statistics from this sample. a) Find the joint pdf of (Y1, .. , Yn). b) Find the joint pdf of (W1, .. , Wn) where W1 = nY1, W2 = (n-1)(Y2 -Y1), W3 = (n - 2)(Y3 - Y2),..., Wn-1 = 2(Yn-1 - Yn-2), Wn = Yn - Yn-1. (c) Show that Wi's are independent and its distribution is identically...

Please answer all the parts neatly with all details. 4. Let X1, X2,... be independent random...

Please answer all the parts neatly with all details. 4. Let X1, X2,... be independent random variables satisfying E(X4) < B for some finite B > 0 and E(Xn)-> . (a) Show that Y = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) < B, E(Y4 (b) Show that for Y, = (Yi +..+ Y)/n, 16B 16B ΣειβΥ< 6B 1 = n4 i=1 6 + n4 ij ΣΕ Υ) E(Y4) n'3 n2 P(Y > €) < oo and...
5. Let X; (i = 1, 2, 3) be be independent gamma random variables with a;...

5. Let X; (i = 1, 2, 3) be be independent gamma random variables with a; = i and B. = 8. a. Find a maximum likelihood estimator of 8 and prove that it is unbiased. b. Show that 2(X1+X2+Xa) is a pivotal quantity for 0. c. Find a 95% confidence interval for 6.
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is...

Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1

15. (30 points) Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 fr...

15. (30 points) Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 from a distribution with p.d.f.f(x) 2x, 0 < x < 1, zero elsewhere. Evaluate E[Yalyj]. [Hint: First find the joint p.d.f. of Y3 and Y4, and then find the conditional p.d.f. of Y4 given Y3 y3] 15. (30 points) Let Y1