Let the pdf of X be defined by f(x) = 4 2 x, 0 < x 2 , 4 2 x + 4 , 2 < x , 0, elsewhere

Let X1, X2, ... , Xn be a random sample from N(, 2), where the mean = is such that < < and 2 is a known positive number. Show that the maximum likelihood estimator for is 6 = X.

A random sample X1, X2, ... , Xn of size n is taken from N(, 2), where the variance = 2 is such that 0 << and is a known real number. Show that the maximum likelihood estimator for is 6 = (1/n) n i=1 (Xi )2 and that this estimator is an unbiased estimator of .

A random sample X1, X2, ... , Xn of size n is taken from a Poisson distribution with a mean of , 0 << . (a) Show that the maximum likelihood estimator for is 6 = X. (b) Let X equal the number of flaws per 100 feet of a used computer tape. Assume that X has a Poisson distribution with a mean of . If 40 observations of X yielded 5 zeros, 7 ones, 12 twos, 9 threes, 5 fours, 1 five, and 1 six, find the maximum likelihood estimate of .

For determining half-lives of radioactive isotopes, it is important to know what the background radiation is in a given detector over a specific period. The following data were taken in a -ray detection experiment over 98 ten-second intervals: 58 50 57 58 64 63 54 64 59 41 43 56 60 50 46 59 54 60 59 60 67 52 65 63 55 61 68 58 63 36 42 54 58 54 40 60 64 56 61 51 48 50 60 42 62 67 58 49 66 58 57 59 52 54 53 53 57 43 73 65 45 43 57 55 73 62 68 55 51 55 53 68 58 53 51 73 44 50 53 62 58 47 63 59 59 56 60 59 50 52 62 51 66 51 56 53 59 57 Assume that these data are observations of a Poisson random variable with mean . (a) Find the values of x and s2. (b) What is the value of the maximum likelihood estimator of ? (c) Is S2 an unbiased estimator of ? (d) Which of x and s2 would you recommend for estimating ? Why? You could compare the variance of X with the variance of S2, which is Var(S2) = (2n + n 1) n(n 1) .

Let X1, X2, ... , Xn be a random sample from distributions with the given probability density functions. In each case, find the maximum likelihood estimator 6. (a) f(x; ) = (1/ 2) x ex/ , 0 < x < , 0 << . (b) f(x; ) = (1/2 3) x2 ex/ , 0 < x < , 0 << . (c) f(x; ) = (1/2) e|x| , < x < , <<. Hint: Finding involves minimizing |xi |, which is a difficult problem. When n = 5, do it for x1 = 6.1, x2 = 1.1, x3 = 3.2, x4 = 0.7, and x5 = 1.7, and you will see the answer. (See also Exercise 2.2-8.)

Find the maximum likelihood estimates for \(\sigma_1 = \mu\) and \(\sigma_2 = \sigma_2\) if a random sample of size 15 from \(N(\mu, \sigma^2)\) yielded the following values: 31.5 36.9 33.8 30.1 33.9 35.2 29.6 34.4 30.5 34.2 31.6 36.7 35.8 34.5 32.7

Let f(x; ) = x1, 0 < x < 1,  = { : 0 < < }. Let X1, X2, ... , Xn denote a random sample of size n from this distribution. (a) Sketch the pdf of X for (i) = 1/2, (ii) = 1, and (iii) = 2. (b) Show that 6 = n/ ln 9n i=1 Xi is the maximum likelihood estimator of . (c) For each of the following three sets of 10 observations from the given distribution, calculate the values of the maximum likelihood estimate and the methodof-moments estimate of : (i) 0.0256 0.3051 0.0278 0.8971 0.0739 0.3191 0.7379 0.3671 0.9763 0.0102 (ii) 0.9960 0.3125 0.4374 0.7464 0.8278 0.9518 0.9924 0.7112 0.2228 0.8609 (iii) 0.4698 0.3675 0.5991 0.9513 0.6049 0.9917 0.1551 0.0710 0.2110 0.2154

Let \(f(x ; \theta)=(1 / \theta) x^{(1-\theta) / \theta}, 0<x<1,0<\theta< \infty\) (a) Show that the maximum likelihood estimator of \(\theta\) is \(\widehat{\theta}=-(1 / n) \sum_{i=1}^{n} \ln X_{i}\). (b) Show that \(E(\widehat{\theta})=\theta\) and thus that \(\widehat{\theta}\) is an unbiased estimator of \(\theta\).

Let \(X_1, X_2, \dots , X_n\) be a random sample of size n from the exponential distribution whose pdf is \(f(x; \theta) = (1/\theta){e?x/\theta}, 0 < x < \infty, 0 <\theta< \infty\). (a) Show that \(\bar X\) is an unbiased estimator of \(\theta\). (b) Show that the variance of \(\bar X\) is \(\theta^2/n\). (c) What is a good estimate of \(\theta\) if a random sample of size 5 yielded the sample values 3.5, 8.1, 0.9, 4.4, and 0.5?

Let \(X_1, X_2, \ldots , X_n\) be a random sample of size n from a geometric distribution for which p is the probability of success. (a) Use the method of moments to find a point estimate for p. (b) Explain intuitively why your estimate makes good sense. (c) Use the following data to give a point estimate of p: 3 34 7 4 19 2 1 19 43 2 22 4 19 11 7 1 2 21 15 16

Let X1, X2, ... , Xn be a random sample from a distribution having finite variance 2. Show that S2 = n i=1 (Xi X)2 n 1 is an unbiased estimator of 2. Hint: Write S2 = 1 n 1  n i=1 X2 i nX2  and compute E(S2).

Let X1, X2, ... , Xn be a random sample from b(1, p) (i.e., n Bernoulli trials). Thus, Y = n i=1 Xi is b(n, p). (a) Show that X = Y/n is an unbiased estimator of p. (b) Show that Var(X) = p(1 p)/n. (c) Show that E[X(1 X)/n] = (n 1)[p(1 p)/n2]. (d) Find the value of c so that cX(1 X) is an unbiased estimator of Var(X) = p(1 p)/n.

Let \(X_1, X_2, \cdots , X_n\) be a random sample from a uniform distribution on the interval \((\theta ? 1, \theta + 1)\). (a) Find the method-of-moments estimator of \(\theta\). (b) Is your estimator in part (a) an unbiased estimator of \(\theta\)? (c) Given the following n = 5 observations of X, give a point estimate of \(\theta\): 6.61 7.70 6.98 8.36 7.26 (d) The method-of-moments estimator actually has greater variance than the maximum likelihood estimator of \(\theta\), namely \([\mathrm{min}(X_i)+\mathrm{max}(X_i)]/2\). Compute the value of the latter estimator for the n = 5 observations in (c).

Let X1, X2, ... , Xn be a random sample of size n from a normal distribution. (a) Show that an unbiased estimator of is cS, where c = n 1  n 1 2  2  n 2  . Hint: Recall that the distribution of (n 1)S2/2 is 2(n1). (b) Find the value of c when n = 5; when n = 6.(c) Graph c as a function of n. What is the limit of c as n increases without bound?

Given the following 25 observations from a gamma distribution with mean \(\mu = \alpha \theta\) and variance \(\sigma^2 = \alpha \theta^2\), use the method-of-moments estimators to find point estimates of \(\alpha\) and \(\theta\): 6.9 7.3 6.7 6.4 6.3 5.9 7.0 7.1 6.5 7.6 7.2 7.1 6.1 7.3 7.6 7.6 6.7 6.3 5.7 6.7 7.5 5.3 5.4 7.4 6.9

An urn contains 64 balls, of which N1 are orange and N2 are blue. A random sample of n = 8 balls is selected from the urn without replacement, and X is equal to the number of orange balls in the sample. This experiment was repeated 30 times (the 8 balls being returned to the urn before each repetition), yielding the following data: 300111131120131 010211232243112 Using these data, guess the value of N1 and give a reason for your guess.

Let the pdf of X be defined by f(x) =  4 2  x, 0 < x 2 ,  4 2  x + 4 , 2 < x , 0, elsewhere, where  = { : 0 < 2}. (a) Sketch the graph of this pdf when = 1/2, = 1, and = 2. (b) Find an estimator of by the method of moments. (c) For the following observations of X, give a point estimate of : 0.3206 0.2408 0.2577 0.3557 0.4188 0.5601 0.0240 0.5422 0.4532 0.5592

Let independent random samples, each of size n, be taken from the k normal distributions with means , j = 1, 2, . . . , k, respectively, and common variance . Find the maximum likelihood estimators of c and d.

Let the independent normal random variables Y1, Y2, ... , Yn have the respective distributions N(, 2x2 i ), i = 1, 2, ... , n, where x1, x2, ... , xn are known but not all the same and no one of which is equal to zero. Find the maximum likelihood estimators for and 2.