Probem.ume that the distribution 4o, Lat bethe sample mean of a random sample of n- 16...
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and variance σ2 16. Find P (49 < Xs <51) and P (49< X <51) 2. Let Y = X1 + X2 + 15 be the sun! of a random sample of size 15 from the population whose + probability density function is given by 0 otherwise 1. Let Xi l be a random sample from a normal distribution with mean μ 50 and...
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and variance σ , with known μ and unknown σ' . You are given that Σ(X-μ)2 is sufficient for σ a) Find El Σ(X-μ). |. Show all steps. Use the fact that: Var(Y)-E(P)-(BY)' i-1 b) Find the MVUE of σ.
16. (10 points) Let X.X.... be iid from distribution with mean pand variance a The sample variance is defined as S2 = 21 A. Prove that E(S2) = y2 for any distribution B. If the distribution is normal, find another way to prove that E(S) = 0 C. If the distribution is normal, find V(S)
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
Assume that we have a finite population of 5 elements on which we measured a random variable X. Below are the observations of Xover the population: {0,1,2,3,4). Assume that we want to select a random sample of size 3 form this population. Let S be the sample variance. a. Find the distribution of S2 b. Calculate E(S) c. Calculate Var(s)
I. Consider a random sample of size n from a normal population with variance σ (n-1s2 (a) Find and identify the distribution of S2. Hint: what do you know about (1) (b) Find EIS] and Var(S2)
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
1. Suppose that {X1, ... , Xn} is a random sample from a normal distribution with mean p and and variance o2. Let sa be the sample variance. We showed in lectures that S2 is an unbiased estimator of o2. (a) Show that S is not an unbiased estimator of o. (b) Find the constant k such that kS is an unbiased estimator of o. Hint. Use a result from Student's Theorem that (n − 1)52 ~ x?(n − 1)...
(2) Let X, X, be a random sample from normal distribution N (,o2), stribution N(u, a and let S2 be the sample variance: (a) [8pts] show that ES-g? (b) [8pts] For a random sample of size 2 (i.e. n 2), derive that /02 ~ Z2 where Z has standard normal distribution.
Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)