Problem H5 Let X be a single observation (n-1) from the following distribution: f(rle)-o elsewhere NOTE:XBeta(0,...
Let f(x, y) = kxy, for 0 <x< 1 and 0 <y<1 and 0 elsewhere, a) Find k b) Find marginal pdfs. c) Are X and Y independent? d) Find P(X<0.5, Y>0.5).
Consider two pdfs f(C) and f, (x) which are defined as follows 1 0sxs1 f(x)=16 otherwise and i(x)- 0 otherwise Suppose that a single observation X is to be taken from f(x) where f(x) is either :G(x) or f(x). The following simple hypotheses are to be tested: Ho:(x)- (x) 4: f (x) = f(x) Determine a test procedure δ for which the value of α (6) + 23(6) is minimum. Report both the rejection region and the corresponding minimum value...
1 1 Let X be a single observation from a population with density function 0-e- for x = 0, 1, 2, ,00 0 otherwise, What is the form likelihood ratio test critical region for testing Ho : θ-2 versus Ha : 1 1 Let X be a single observation from a population with density function 0-e- for x = 0, 1, 2, ,00 0 otherwise, What is the form likelihood ratio test critical region for testing Ho : θ-2 versus...
2. (17 points total) Let X be a random variable with the following distribution: f(x) 2a 0 otherwise Consider the following set of hypotheses: Ho:Os 4 and H4 :0>4. You observe one data point x and decide to reject the null hypothesis if x e. (Contributed by XX.) a) (6 pts) Find your probability of Type I error. b) (6 pts) Consider the altemative hypothesis H: θ-5. find your probability of Type II error Given the rejection region in part...
For f(x, y) = k(x2 + y2), 0<x< 1 and 0 <y<1 and 0 elsewhere: a) Find k. b) Are X and Y independent? c) Find P(X<0.5, Y>0.5), P( X = 0.5, Y>0.5).
Problem 3. The random variable X has density function f given by 0, elsewhere (a) Assuming that 6 0.8, determine K (b) Find Fx(t), the c.d.f. of X (c) Calculate P(0.4 <X < 0.8)
Let X1, . . . , Xn ∼ Geo(θ), f(x)= θ(1-θ)^x, and we wish to test H0 : θ ≤ 1/3 vs H1 : θ > 1/3. a) Using the full sample, X1....Xn, find the form of the UMP test for the hypotheses H0: θ=1/3 vs H1: θ=1/2. b)If n=15 and α = 0.1, what is the rejection region and the size of test in (a)?
Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question: Fact 1: If X EXP(0) then 2X/0~x(2). Fact 2: If V V, are a random sample from a x2(k) distribution then V V (nk) (a) Suppose that we wish to test Ho : 0 against H : 0 = 0, where 01 is specified and 0, > Oo. Show that the likelihood ratio statistic AE, O0,0)f(E)/ f (x;0,)...
Suppose a single measurement is taken from a distribution with pdf f(x)=?e−?x, x>0. The hypotheses are H0∶?=1 versus HA∶?<1, and the null hypothesis is rejected if x ≥ 3.2. a) Calculate the probability of committing a type I error. b) Calculate the probability of committing a type II error if, in fact, ? = 1∕5.
ONLY A) B) D) 4 Let X be a single observation from the density f(x; 0)= Ox® -110, 1)(x), where 0 >0. (a) In testing Ho: 0 <1 versus H 1:8 > 1, find the power function and size of the test given by the following: Reject H , if and only if X > . (6) Find a most powerful size-a test of Ho:8=2 versus H 1:0= 1. (c) For the loss function given by [(do; 2) = f(d1;...