1. Let X be an iid sample of size n from a continuous distribution with mean /i, variance a2 and such that Xi e [0, 1] for all i e {1,...,n}. Let X = average. For a E (0,1), we wish to obtain a number q > 0 such that: (1/n) Xi be the sample Р(X € |и — 9. и + q) predict with probability approximately In other words, we wish to sample of size n, the average X...
Practice: Compute the Kolmogorov-Smirnov Test Statistic 1 point possible (graded) Let X1, ..., Xn be iid samples with cdf F, and let F° denote the cdf of Unif(0,1). Recall that F° (t) = t. 1(t € [0,1]) +1.1(t > 1). We want to use goodness of fit testing to determine whether or not X1,...,x, iid Unif(0,1). To do so, we will test between the hypotheses = 70 H : F(t) H :F(t) + F. To make computation of the test...
Let X,, X,,... be independent and identically distributed (iid) with E X]< co. Let So 0, S,X, n 2 1 The process (S., n 0 is called a random walk process. ΣΧ be a random walk and let λ, i > 0, denote the probability 7.13. Let S," that a ladder height equals i-that is, λ,-Pfirst positive value of S" equals i]. (a) Show that if q, then λ¡ satisfies (b) If P(X = j)-%, j =-2,-1, 0, 1, 2,...
Exercice 5. Let Xi, ,Xn be iid normal randon variables : Xi ~ N(μ, σ2). We denote 4 Tl Show that (İ) ils2 (i.e., that x is independent of 82). (ii) x ~ N(μ, σ2/n). (iii) !뷰 ~ เลี้-1
Suppose Xi, X2, ,Xn is an iid N(μ, c2μ2 sample, where c2 is known. Let μ and μ denote the method of moments and maximum likelihood estimators of μ, respectively. (a) Show that ~ X and μ where ma = n-1 Σηι X? is the second sample (uncentered) moment. (b) Prove that both estimators μ and μ are consistent estimators. (c) Show that v n(μ-μ)-> N(0, σ ) and yM(^-μ)-+ N(0, σ ). Calculate σ and σ . Which estimator...
Let X, Y be iid random variables that are both uniformly distributed over the interval (0,1). Let U = X/Y. Calculate both the CDF and the pdf of U, and draw graphs of both functions.
Let X1, X2,· · ·iid B(1, x), i.e,P(X1= 1) =x= 1−P(X1= 0), where x∈ [0,1]. Let Sn = X1+X2+· · ·+Xn. What can you say about the limiting behaviour of Sn/n from strong law large number
11.3.4 Let the random variables Xil, ,Xini be iid Nund2), i 1,2, and that the X,'s be independent of the X2j's. Here we assume that , μ2, σ are all unknown and (μι, μ2) E R × R, σ E R+. with preassigned α E (0,1) and a real number D, show that a level α LR test for Ho : μι-μ,-D versus Hi : μ,-μ, D would reject Ho if and only if {n-1 + 7DSptnitn-2,a/2. {Hint: Repeat the...
6. Let X1, . . . , Xn denote a random sample (iid.) of size n from some distribution with unknown μ and σ2-25. Also let X-(1/ . (a) If the sample size n 64, compute the approximate probability that the sample mean X n) Σηι Xi denote the sample mean will be within 0.5 units of the unknown p. (b) If the sample size n must be chosen such that the probability is at least 0.95 that the sample...
2. Let X1, X2, X3 ..., X, be iid b(1, p) random variables. Let Sn = 27-1Xthen prove that Sn-E(Sn) N(0,1) as n +00. (Sn)