Set α >1. Tahe Xt> , and define a rove b)rove thet Xe X4Xa c)Trove that lim Xn-
Set α >1. Tahe Xt> , and define a rove b)rove thet Xe X4Xa c)Trove that lim Xn-
We define the set X ⊆ L^∞ by X = { {xn} ∈ L^∞ : lim n→∞ xn = 1 } Prove that the set X with the subspace metric d∞|X×X is a complete metric space.
The sample data x1,x2,...,xn sometimes represents a time series, where xt = the observed value of a response variable x at time t. Often the observed series shows a great deal of random variation, which makes it difficult to study longer-term behavior. In such situations, it is desirable to produce a smoothed version of the series. One technique for doing so involves exponential smoothing. The value of a smoothing constant α is chosen (0 < α < 1). Then with...
x-4 a) lim x-2 X-2 xe* b) lim *-01-et lim (1+x)'* X+00
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
Assume that V is the set of all complex sequences, (xn), that satisfy the relation Xn+nXn+1 – ixn+4 = 0 for all n E N. Furthermore, assume that F = C and for a E C, (2n), (yn) € V define (xn) + (yn) = (xn + yn), a(xn) = (axn) Is V a vector space over C? Justify your answer.
19. A cone in R^ is a set C such that if xe C, then Àxe C for all scalars 1>0. A finite cone satisfies the definition (3). Define a cone in R’ that isn't a finite cone.
Suppose that x1, . . . , xn are a random sample from a B(α, β) distribution: f(x; α, β) = x^(α-1) (1-x)^(β-1) Here E[X] = α/(α + β) and E[X^2 ] = ((α + 1)α)/{(α + β + 1)(α + β)}. (a) Show that the method of moments, using the first two moments, gives the equations 0 = α(1 − m1 ) − βm1 m1 − m2 = α(m2 − m1 ) + βm2 (b) Determine the method of moments...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...
(1) Let C[a, b] denote the set of functions that are continuous on (a,b). Define a function on pairs of function fig E C[a, b] by dı(,9) = /\f(x) – 9(2)| dx. Prove that (C[a,b], du) is a metric space.