Let X, X2, ..., X, be independent with X-Gamma (a,b). Let Y = EX. Prove that Y-Gamma (a,b)
Problem 5. Let a E Z. Prove that if 2 la, then ged (a, a 2)
1. Let H- ta + bija, b e R, ab 20). Prove or disaprove that H is a subgroup of C under addition. 2. Let a and b be elements of an Abelian group and let n be any integer. Prove that (ab)"- a
Exercise 6 requires using Exercises 4 and 5.
Exercise 4. Let a be any real number. Prove that the Euclidean translation Ta given by Ta(x, y)(a, y) is a hyperbolic rigid motion. *Exercise 5. Let a be a positive real number. Prove that the transformation fa: HH given by fa(x, y) (ax, ay) is a hyperbolic rigid motion Exercise 6. Prove that given any two points P and Q in H, there exists a hyperbolic rigid motion f with f(P)...
Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω R b probability space (2, F, P) with the gamma distribution Ta,n. Does there exist a random variable e a random variable on a
Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω R b probability space (2, F, P) with the gamma distribution Ta,n. Does there exist a random variable e a random variable...
Let X have a gamma distribution with parameters a > 2 and 3 > 0. (a) Prove that the mean of 1/X is B/(a - 1). (b) Prove that the variance of 1/X is 82/[(a - 1)(a 2)].
5-4. Prove that the MGF of Gamma distribution is В f(t) В — t. 5-5. Let X1, X2,... ,X be independent with X~ Gamma (ai,ß). Let Y = EX4. Prove that Y Gamma (Eai,ß)
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.
Below are sample questions: [5] 6. Let X ~ GAMMA(0,k). Prove that Y = 2x 2K)
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval [a, b]....