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An Introduction to Mathematical Statistics and Its Applications (6th Edition)
- Consider a random sample X1, . . . , Xn of size n ≥ 2 from the normal distribution N (0, σ2 ) with parameter σ2 > 0 Suppose that the prior distribution for σ2 is inverse Gamma with densityarrow_forwardSuppose that X1, . . . , Xn is a random sample from the Normal distribution N (0, σ2 ) with parameter σ > 0, Find the Maximum Likelihood Estimation of σ.arrow_forwardSuppose that X | N ∼ Binom(N, p), and that N is also a random variable with distributionBinom(m, r). Find the marginal distribution of Xarrow_forward
- The weight of breads prepared in a baker is randomly distributed with a mean of 400 grams and standarddeviation of 16 grams. Using an independently selected sample of 16 breads,finda) the mean and standard deviation of the average weight of these 16 samples.b) using central limit theorem, estimate the probability that the average weight of samples is less than390.arrow_forwardAccording to a study, the length of pregnancy has an unknown distribution with a mean of 265 days and a standard deviation of 15.4 days. Let XX be the length of pregnancy for a randomly selected female and let ¯¯¯XX¯ be the average length of pregnancy for a random sample of size 22. 1. Describe the probability distribution of XX and state its parameters μμ and σσ: X∼X∼ ( μ=μ= , σ=σ= ) 2. Explain why the Central Limit Theorem can or can not be applied The Central Limit Theorem for Sample Means be applied becausearrow_forwardA computer of random numbers with a uniform distribution between 0 and 1 manufactures. A standard deviation of 1 by adding the n random numbers generated. Normal distribution is desired. Where the found n is large enough assuming that the new random variable generated from the sum of these n random numbers what is the average?arrow_forward
- Suppose the traffic light at a specific intersection changes to green at 4-minute intervals. If a car arrives at the traffic light at a random time, then X = waiting time in minutes, is uniformly distributed between 0 and 4. Suppose a random sample of 35 cars are observed and their waiting time is recorded. What is the probability that the sample mean of the waiting times is less than 1.5 minutes?arrow_forwardSuppose you sample repeatedly from the continuous uniform distribution on [0, 1], until you get a number that is less than your previous sample. What is the expected number of samples you obtain?arrow_forwardA random group of students was selected from a large student conference to analyze their class in school. Is there evidence to reject the hypothesis that the number of students is equally distributed between the four classes , = 0.052arrow_forward
- Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.arrow_forwardLet X be a random variable with a uniform distribution over the interval [-2 , 4]. Determine the standard deviation of X.arrow_forwardA new printing machine is tested about the number of printing errors per 5m2 paper. It is assumed that the number of printing errors will have Poisson distribution with parameter l. a) If X1, X2, . . . , Xn is a random sample of size n, estimate the parameter of the distribution. b) The randomly selected 12 papers are inspected, and the number of printing errors are found as 2, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, and 0. Estimate the mean number of printing errors, and write down the distribution function. c) In order to estimate the distribution parameter with 0.3 error and 4% risk, find the minimum sample size.arrow_forward
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