1. Let X1,..., X, be an iid sample from Bernoulli(p). (a) Find the method of moment estimator of p. (b) Find the MLE for p.
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- Consider a random sample from NB(r, p) where the parameter r is known to be 3. An experiment is run with n = 10 trials and the sample mean is observed to be x̄ = 0.6. (a) Derive a formula for the MLE p̂ as a function of n, r and X̄ . (b) Find the estimate of p. (c) Find the MLE for the population mean.Let X1, . . . , Xn be iid from Exp(β). Find the distribution of the sample maximum Y = X(n). Is this an exponential distribution? (This question was rejected this morning. I did double check and there is no missing info with this question.)EX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.
- Let Y, represent the ith normal population with unknown mean 44, and unknown variance of for i 1,2. Consider independent random samples, Ya, Y₁2,,Yin, of size ni, from the ith population with sample mean Y, and sample variance S?=1Σj=1(Y₁j - Y₁². (a) What is the distribution of Y;? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho : μi = μio versus Ha: Pipio when of is unknown and n, is small. (c) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a.3. Let X1,.., X,, be a random sample from f(r, 0) = , 0 0. Let Y = min;{X,}. Find consistent estimators of 20 +4 and 502 + e° based on Y. %3D(11.)Perform the test of hypotheses indicated, using the data from independent samples given. Use the p value approach. (The p-value can be only approximated.) a. Test Ho : H - Hz = 50 vs. Ha : µ, - u, > 50 @a = 0.01: = lu n1 = 30, x1 = 681 s1 = 8 n2 = 27, T2 = 625, 82 = 8 %3D b. Test Ho : 41 – Hz = 35 vs. Ha : Hi – Hy # 35@a = 0.10: n1 = 36, T1 = 325, 81 = 11 n2 = 29, T2 = 286. s2 = 7
- Calculate value of (t) if: first sample mean = 85, second sample mean = 91, and o x1-x2 = 3.Let X1, X2, ..., X, be a random sample from a normal distribution with mean µ and variance o?. (a) Find an unbiased estimator for o?. ( b) Show that Σ(x-Χ? Σ (Χ )- nX. i=1 i=1 (c) Find an estimator for o? using the method of moments. Is this estimator bi- ased/unbiased? Is it consistent?8. Let y be a normal random variable with a constant mean E(y) = 4, constant variance var(y.) = o², and a covariance cov(y, y;) = 0 for t# j. Consider the sample mean j = E %3D A. Show that the variance of the sample mean y is .
- Let I be an indicator variable with p(I) = {p , I=1 } {1-p , I=0}. Show that the variance of I is p(1-p)B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.I need the answer quickly