Let 1 3 p S q and suppose that for 1 Kjs n we have j2 0. Prove that 1 1 np i-1
Let 1 ≤ p ≤ q and suppose that for 1 ≤ j ≤ n we have xj ≥ 0. Prove that ( n ∑ j=1 (xj^q) )^ (1/q) ≤ ( n ∑ j=1 (xj^p) )^ (1/p) ≤ n^ (1/p - 1/q) ( n ∑ j=1 (xj^q) )^ (1/q)
Let z, y > 0. Prove that for any ε > 0 and p, q > 1 so that = 1 we have q + is 1s known as Young's inequality with E. Let z, y > 0. Prove that for any ε > 0 and p, q > 1 so that = 1 we have q + is 1s known as Young's inequality with E.
+o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1 +o0 P(A,) 0(n N4 0, 2. Let A A = Q , prove i1
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics. 3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
Please help me solve 3,4,5 3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
A2 Let X B(n,p) with known n. Then E(X) np and Var (X) np(1- p). Let p X be an estimator of p. a. If n is large (large enough np> 10 and n(1 - p)> 10), what is the (approximate) distri- bution of p? b. We talked in class that providing a confidence interval is "better" than a point esti- mate. Suppose X = 247 (247 successes) is observed in B(450, p) experiment. Suggest a 95% confidence interval for...
(n) 6. Let X ~ Binomial (n,p). Prove that a. Ex=0 (6)p*(1 – p)n-* = ... = 1 b. E[X] = 21-0 x()p*(1 - 2)^-^ = = mp c. Var[X] = x=0x2 (1)p*(1 – p)n-x – (np)2 = ... = np(1 – p) d. My(t) = ... = (pet + 1 - p)n
Suppose we are interested in estimating the proportion p of a population that has a certain disease. As in Section 2.3 let y;-1 if person i has the disease, and yi 0 if person i does not have the disease. Then p . a Show, using the definition in (2.13), that 22 N- np If the population is large and the sampling fraction is small, so that write (2.26) in terms of the CV for a sample of size 1....
Let P be some probability measure on sample space S = [0, 1]. (a) Prove that we must have limn→∞ P((0, 1/n) = 0. (b) Show by example that we might have limn→∞ P ([0, 1/n)) > 0.
Let p and n be integers. Prove that, if p is prime, then gcd(p, n) = p or gcd(p, n) = 1. . . (i.) Using proof by contrapositive (ii.) Using proof by contradiction