10. Let X1, ..., Xn be i.i.d. from some distribution Qo, and let X = (X1+ ... + Xn)/n be the sample average. (a) Show that S² = E(X; – X)²/(n – 1) is unbiased for o? = = Varø(X;). | (b) If Qo is the Bernoulli distribution with success probability 0, then a unique best unbiased estimator for o? can be obtained by calculating E[ô²|X], where ở² is some unbiased estimator. Find this unique best unbiased estimator. (c) If Qo is the exponential distribution with failure rate 0, compute an unbiased esti- mator ô? for o² = 1/0² in terms of X. (d) Again, consider the scenario given in part (c), where the calculation of the unique best unbiased estimator using E[ô²\X] also applies. Give a formula for E[X?|X = c].

10. Let X1, ..., Xn be i.i.d. from some distribution Qo, and let X = (X1+ ... + Xn)/n be the sample average. (a) Show that S² = E(X; – X)²/(n – 1) is unbiased for o? = = Varø(X;). | (b) If Qo is the Bernoulli distribution with success probability 0, then a unique best unbiased estimator for o? can be obtained by calculating E[ô²|X], where ở² is some unbiased estimator. Find this unique best unbiased estimator. (c) If Qo is the exponential distribution with failure rate 0, compute an unbiased esti- mator ô? for o² = 1/0² in terms of X. (d) Again, consider the scenario given in part (c), where the calculation of the unique best unbiased estimator using E[ô²\X] also applies. Give a formula for E[X?|X = c].

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Similar questions

Suppose (S, P) is a binomial distribution with n trials and probability of success p. Let X be the random variable X(k) = k³, where k is the number of successes. Calculate E[X].
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
Q1. let Y₁ < Y₂ < Y3 < Y4 < Y5 are the order statistics of the random sample of size 5 from the distribution : f(x) = 3x², 0
Suppose X is a normal random variable with mean u = 17.5 and o = 6. A random sample of size n = 24 is selected from this population. a. Find the distribution of b. Find P(X < 14) and P(X < 14) Find P(15
1. A discrete random variable X follows the Uniform distribution if X takes values x = 1, 2, .., N, with P(X = x) = 1/N. Compute E(X), E(X²) and the variance Var(X). You may use the following identities: п(п + 1) (1) 2 п(п + 1)(2n + 1) (2) i=1
(a) Let Y be a random variable distributed as X. Determine E(Y) in terms of r. (b) Let {X1, X2, . .. , Xn} be a random sample drawn from a normal distirbution with mean u and 1 variance o?. Denote S E-(X; – X)² as the sample standard deviation. Use the 1 n - result in part (a), or otherwise, to find E(S). (c) Find an unbiased estimator for the population standard deviation o.
4.2 The probability distribution of the discrete random variable X is f(x) = 3 (*)*** Find the mean of X. 3-x (¹)ª (²³)³—ª, x = 0, 1, 2, 3.
The profits of a mobile company are normally distributed with Mean of R.O (180) and standard deviatic of R.O (8). a. Find the probability that a randomly selected mobile has a profit greaterthan R.O ( 190). b. Any mobile phone which profit is greater than R.O (190) is defined as expensive. Find the probability that a randomly selected mobile has aprofit greaterthan R.O ( (200) given that it is expensive. Half of expensivemobile phones have aprofit greaterthan R.O h. Find the value of h. C.
#4: Suppose that X1,... , X7n is an i.i.d. random sample drawn from a U[a, 1] uniform distribution, where 0a < 1. Suppose that you want to find an unbiased estimator, â, for a. (a) By definition, what relation must be satisfied for â to be an unbiased estimator of a (b) Use the first-order statistic to find an unbiased estimator of a. Recall that the first-order statistic has the p.d.f: п-1 g(yn)nf(yn) f(x)da Уп where f is the p.d.f. of each of the X;'s. (c) Find an alternative unbiased estimator â of a, which is still based on the random sample (hint: think of a common statistic we calculate from the random sample)
Let X1, X2......,X100 be independent binomial random variables with parameter p=0.25. The sample mean is : (First image) a. Find the value of E(X) and Var(X). b. State the sampling distribution of X and explain why has the sampling distribution stated. c. Find P(X<25.5). d. Find (second image)
a and b