50] 1. Suppose that Xi,X2.. are independent and identically distributed Bernoulli random vari-abl...
Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables with common probability function ()p(1-p) m m-a ; x 0,1,. ., m otherwise 0 where m is known and p is unknown (a) Obtain the Sequential Probability Ratio Test of Ho p = po versus HA p P, where pi > po, with significance level 0.01 and power 0.95. Describe the test precisely; (b) For the case where po 3/8,pı = 1/2, m =...
valu Exercises 8.2. x, . . . ,x, nd G(p), the geometric distribution with mean 1/p. Assume that e size n is sufficiently large to warrant to invocation of the Central Limit Theo- . Suppose se that Xi , . . . X, Use the asymptotic d confidence interval for p Suppose that XN(0, o2) (a) Obtain the asymptotic distribution of the second istribution of p 1/X to obtain an approximate 100(1-u)% Suppose sample moment m2 -(I/n)i X. (b) Identify...
Question 3: Bernoulli distribution (23/100 points) Consider a random sample X1,...,Xn from a Bernoulli distribution with unknown parameter p that describes the probability that Xi is equal to 1. That is, Bernoulli(p), i = 1, ..., n. (10) The maximum likelihood (ML) estimator for p is given by ÔML = x (11) n It holds that NPML BIN(n,p). (12) 3.a) (1 point) Give the conservative 100(1 – a)% two-sided equal-tailed confidence interval for p based on ÔML for a given...
Problem 7. Let Xi, X2,..., Xn be i.i.d. (independent and identically distributed) random variables with unknown mean μ and variance σ2. In order to estimate μ and σ from the data we consider the follwing estimates n 1 Show that both these estimates are unbiased. That is, show that E(A)--μ and
(a) Suppose that Xi, X2,... are independent and identically distributed random variables each taking the value 1 with probability p and the value-1 with probability 1-p For n 1,2,..., define Yn -X1 + X2+ ...+Xn. Is {Yn) a Markov chain? If so, write down its state space and transition probability matrix. (b) Let Xı, X2, ues on [0,1,2,...) with probabilities pi-P(X5 Yn - min(X1, X2,.. .,Xn). Is {Yn) a Markov chain and transition probability matrix. be independent and identically distributed...
1. Let Xi,X2,... .X, be independent identically distributed N( 5,4 ran dom variables Find (use tables or software) t such that (a) P(X-51 > t) = .95 1-2-) >t) = .05 (c) P0% (X (d) P(LL52 *)2 > t) > t) .95 .95 where 82- Σ(x,-x)2
1. The random variables Xi, X2,.. are independent and identically distributed (iid), each with pdf f given in Assignment 4, Question 1. Let Sn- Xi+.+X Using the Central Limit Theorem and the graph of the standard normal distribution in Figure 1, approximate the probability P(S100 >600). Express your answer in the format x.x-10-x. Verify your answer by simulating 10,000 outcomes of Si00 and counting how many of them are > 600. Show the code 1.00 0.95 0.90 0.85 1.2 1.4...
Let xi, 1-1,2 1,50 be independent random variables each being uniformly distributed over the interval (0.1) Find the approximate value of P{ EX; 30} You may use the fact that %10) - 0.9928. Lang! 0:00 717 { Hint EXiS is a sequence of unfoomly distributed condom va table with meon V and varionce a2 then n Vn L e follows standard no mal distributions
1. The random variables Xi, X2,... are independent and identically distributed (iid), . .. are independent and identica each with pdf f given in Assignment 4, Question 1. Let s, X1 + . .. + Xn. Using the Central Limit Theorem and the graph of the standard normal distribution in Figure 1, approximate the probability P(S100 > 600). Express your answer in the format x.x - 10*. Verify your answer by simulating 10,000 outcomes of S1o0 and counting how many...
Independent random samples X1, X2, . . . , Xn are from exponential distribution with pdfs , xi > 0, where λ is fixed but unknown. Let . Here we have a relative large sample size n = 100. (ii) Notice that the population mean here is µ = E(X1) = 1/λ , population variance σ^2 = Var(X1) = 1/λ^2 is unknown. Assume the sample standard deviation s = 10, sample average = 5, construct a 95% large-sample approximate confidence...