An Introduction to Mathematical Statistics and Its Applications (6th Edition)
6th Edition
ISBN: 9780134114217
Author: Richard J. Larsen, Morris L. Marx
Publisher: PEARSON
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Textbook Question
Chapter 5.6, Problem 8Q
Suppose a random sample of size
Find a sufficient statistic for
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Q/Fit (y = ax² + (b/x)) to the following data: (x:1, 2, 3, 4), (y: -1.51, 0.99, 3.88, 7.66). *
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Let X1, . . . , Xn be an iid sample from f(x | θ) = θxθ−1 , 0 < x < 1,
where the parameter θ is positive.
Find the MLE and MOM estimators for θ.
Chapter 5 Solutions
An Introduction to Mathematical Statistics and Its Applications (6th Edition)
Ch. 5.2 - A random sample of size...Ch. 5.2 - The number of red chips and white chips in an urn...Ch. 5.2 - Use the sample y1=8.2,y2=9.1,y3=10.6, and y4=4.9...Ch. 5.2 - Suppose a random sample of size n is drawn from...Ch. 5.2 - Given that y1=2.3,y2=1.9, and y3=4.6 is a random...Ch. 5.2 - Use the method of maximum likelihood to estimate ...Ch. 5.2 - An engineer is creating a project scheduling...Ch. 5.2 - The following data show the number of occupants in...Ch. 5.2 - For the Major League Baseball seasons from 1950...Ch. 5.2 - (a) Based on the random sample...
Ch. 5.2 - Find the maximum likelihood estimate for in the...Ch. 5.2 - A random sample of size n is taken from the pdf...Ch. 5.2 - If the random variable Y denotes an individuals...Ch. 5.2 - For the negative binomial pdf...Ch. 5.2 - The exponential pdf is a measure of lifetimes of...Ch. 5.2 - Suppose a random sample of size n is drawn from a...Ch. 5.2 - Let y1,y2,...,yn be a random sample of size n from...Ch. 5.2 - Prob. 18QCh. 5.2 - A criminologist is searching through FBI files to...Ch. 5.2 - Prob. 20QCh. 5.2 - Suppose that Y1=8.3,Y2=4.9,Y3=2.6, and Y4=6.5 is a...Ch. 5.2 - Find a formula for the method of moments estimate...Ch. 5.2 - Calculate the method of moments estimate for the...Ch. 5.2 - Find the method of moments estimates for and 2,...Ch. 5.2 - Use the method of moments to derive estimates for...Ch. 5.2 - Bird songs can be characterized by the number of...Ch. 5.2 - Prob. 27QCh. 5.3 - A commonly used IQ test is scaled to have a mean...Ch. 5.3 - The production of a nationally marketed detergent...Ch. 5.3 - Mercury pollution is widely recognized as a...Ch. 5.3 - A physician who has a group of thirty-eight female...Ch. 5.3 - Suppose a sample of size n is to be drawn from a...Ch. 5.3 - What confidence would be associated with each of...Ch. 5.3 - Five independent samples, each of size n, are to...Ch. 5.3 - Suppose that y1,y2,...,yn is a random sample of...Ch. 5.3 - If the standard deviation () associated with the...Ch. 5.3 - In 1927, the year he hit sixty home runs, Babe...Ch. 5.3 - A thirty-second commercial break during the...Ch. 5.3 - During one of the first beer wars in the early...Ch. 5.3 - The Pew Research Center did a survey of 2253...Ch. 5.3 - If (0.57,0.63) is a 50% confidence interval for p,...Ch. 5.3 - Suppose a coin is to be tossed n times for the...Ch. 5.3 - On the morning of November 9, 1994the day after...Ch. 5.3 - Which of the following two intervals has the...Ch. 5.3 - Prob. 18QCh. 5.3 - Prob. 19QCh. 5.3 - Prob. 20QCh. 5.3 - Prob. 21QCh. 5.3 - A public health official is planning for the...Ch. 5.3 - Prob. 23QCh. 5.3 - Given that a political poll shows that 52% of the...Ch. 5.3 - Prob. 25QCh. 5.3 - Suppose that p is to be estimated by Xn and we are...Ch. 5.3 - Let p denote the true proportion of college...Ch. 5.3 - Prob. 28QCh. 5.4 - Two chips are drawn without replacement from an...Ch. 5.4 - Suppose a random sample of size n=6 is drawn from...Ch. 5.4 - Prob. 3QCh. 5.4 - A sample of size n=16 is drawn from a normal...Ch. 5.4 - Suppose X1,X2,...,Xn is a random sample of size n...Ch. 5.4 - Prob. 6QCh. 5.4 - Let Y be the random variable described in Example...Ch. 5.4 - Suppose that 14, 10, 18, and 21 constitute a...Ch. 5.4 - A random sample of size 2, Y1 and Y2, is drawn...Ch. 5.4 - A sample of size 1 is drawn from the uniform pdf...Ch. 5.4 - Suppose that W is an unbiased estimator for . Can...Ch. 5.4 - We showed in Example 5.4.4 that 2=1ni=1n(YiY)2 is...Ch. 5.4 - As an alternative to imposing unbiasedness, an...Ch. 5.4 - Let Y1,Y2,...,Yn be a random sample of size n from...Ch. 5.4 - An estimator n=h(W1,...,Wn) is said to be...Ch. 5.4 - Is the maximum likelihood estimator for 2 in a...Ch. 5.4 - Let X1,X2,...,Xn denote the outcomes of a series...Ch. 5.4 - Suppose that n=5 observations are taken from the...Ch. 5.4 - Let Y1,Y2,...,Yn be a random sample of size n from...Ch. 5.4 - Given a random sample of size n from a Poisson...Ch. 5.4 - If Y1,Y2,...,Yn are random observations from a...Ch. 5.4 - Suppose that W1 is a random variable with mean ...Ch. 5.5 - Let Y1,Y2,...,Yn be a random sample from...Ch. 5.5 - Let X1,X2,...,Xn be a random sample of size n from...Ch. 5.5 - Suppose a random sample of size n is taken from a...Ch. 5.5 - Let Y1,Y2,...,Yn be a random sample from the...Ch. 5.5 - Prob. 5QCh. 5.5 - Let Y1,Y2,...,Yn be a random sample of size n from...Ch. 5.5 - Prove the equivalence of the two forms given for...Ch. 5.6 - Let X1,X2,...,Xn be a random sample of size n from...Ch. 5.6 - Let X1,X2, and X3 be a set of three independent...Ch. 5.6 - If is sufficient for , show that any one-to-one...Ch. 5.6 - Show that 2=i=1nYi2 is sufficient for 2 if...Ch. 5.6 - Let Y1,Y2,...,Yn be a random sample of size n from...Ch. 5.6 - Let Y1,Y2,...,Yn be a random sample of size n from...Ch. 5.6 - Suppose a random sample of size n is drawn from...Ch. 5.6 - Suppose a random sample of size n is drawn from...Ch. 5.6 - Prob. 9QCh. 5.6 - Prob. 10QCh. 5.6 - Prob. 11QCh. 5.7 - How large a sample must be taken from a normal pdf...Ch. 5.7 - Let Y1,Y2,...,Yn be a random sample of size n from...Ch. 5.7 - Suppose Y1,Y2,...,Yn is a random sample from the...Ch. 5.7 - An estimator n is said to be squared-error...Ch. 5.7 - Suppose n=Ymax is to be used as an estimator for...Ch. 5.7 - Prob. 6QCh. 5.8 - Prob. 1QCh. 5.8 - Find the squared-error loss [L(,)=()2] Bayes...Ch. 5.8 - Prob. 3QCh. 5.8 - Prob. 4QCh. 5.8 - Prob. 5QCh. 5.8 - Suppose that Y is a gamma random variable with...Ch. 5.8 - Prob. 7QCh. 5.8 - Find the squared-error loss Bayes estimate for in...Ch. 5.8 - Prob. 9Q
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- Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.arrow_forwardThe following table provides values of the function f(x,y). However, because of potential; errors in measurement, the functional values may be slightly inaccurately. Using the statistical package included with a graphical calculator or spreadsheet and critical thinking skills, find the function f(x,y)=a+bx+cy that best estimate the table where a, b and c are integers. Hint: Do a linear regression on each column with the value of y fixed and then use these four regression equations to determine the coefficient c. x y 0 1 2 3 0 4.02 7.04 9.98 13.00 1 6.01 9.06 11.98 14.96 2 7.99 10.95 14.02 17.09 3 9.99 13.01 16.01 19.02arrow_forwardAssume X_1, X_2, .....X_n are random samples from X~Exponential(θ).1) Find the MLE estimator of θ .arrow_forward
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