Suppose the reaction temperature X (in °C) in a certain chemical process has a uniform distribution with A = −6 and B = 6. (a) Compute P(X < 0) (b) Compute P(−3 < X < 3). (c) Compute P(−4≤ X ≤ 5). (Round your answer to two decimal places.)
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Suppose the reaction temperature X (in °C) in a certain chemical process has a uniform distribution with A = −6 and B = 6.
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- 1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?12. Suppose the PDF is given by f(x) = Cx² for 1 ≤ x ≤ 2, and zero otherwise. (a) Find the value of C. (b) Find the likelihood that a point selected at random with respect to this PDF is between 1.5 and 2. (c) With X (x) = x, find E(X) and Var(X).(c) What is the asymptotic distribution of √n(0-0)? 6.2.9. If X1, X2,..., Xn is a random sample from a distribution with pdf ={ f(x; 0) = 303 (x+0)4 0 07. a) Suppose that X is a uniform continuous random variable where 0The inhabitants of a city develop skin cancer at an approximate rate A. For those people who have developed skin cancer, some proportion p E (0, 1) will die from the disease. Assume a simple model such that n, the number of people who develop skin cancer, is distributed Poisson(A). Let X denote the people who die from skin cancer in this city. Then, assuming that every cancer patient is independent of the others, and that the proportion p is constant: n ~ Poisson(X) X\n - Binomial(n, p) Question 5: What is the distribution of n X? n – X|X ~ Binomial X(1-p) 1- p We do not have enough information to answer this question. n|X Poisson(Xp) On - X|X ~ Poisson(A(1 – p) 1 for n > x n|X ~ fn|x (n|x) = n! 1- Di-o i! 0.w.(d) P(|Z|>0.7) = (Round to four decimal places as needed.) (e) P(- 1.1sZs1.2) = (Round to four decimal places as needed.)Suppose the reaction temperature X (in °C) in a certain chemical process has a uniform distribution with A = −6 and B = 6. (d) For k satisfying −6 < k < k + 4 < 6, compute P(k < X < k + 4). (Round your answer to two decimal places.)(a) Let X be a normal r.v. with mean u =12 and standard deviation o = 1.5. Find P( X 13 ) = (c) Let X be a normal r.v. with mean u =7 and standard deviation o = 1.5. Find P( 5Please show as many steps as you can so i can understand the proccess.The inhabitants of a city develop skin cancer at an approximate rate X. For those people who have developed skin cancer, some proportion p E (0, 1) will die from the disease. Assume a simple model such that n, the number of people who develop skin cancer, is distributed Poisson(A). Let X denote the people who die from skin cancer in this city. Then, assuming that every cancer patient is independent of the others, and that the proportion p is constant: ~ U Poisson(A) X\n - Binomial(n, p) Question 4: What is the marginal distribution of X? Hint: two primary ways to do this: 1. you can summate out n from the joint distribution directly (more straightforward, but tricky algebra) o Pay attention to the lower limit of n o Remember that i=0 2. use MGFS + iterated expectation: E[etx] = En|Exn[etx n|| and then recognize the distribution corresponding to the MGF (need to understand MGFS and iterated expectation). o Obtain the inner expectation by using the Binomial theorem. O X Binoтial(n, Ap)…Customers arrive at a facility according to a Poisson process N(t) of rate λ = 4.5 customers/hour. Each customer is admitted to the facility with probability p = 0.8. All customers, who are not admitted, leave and do not come back. Let X(t) be the number of customers admitted from time 0 to time t. Compute the following: (a) The mean value E[X(6.5)] = 23.4 (b) The probability P(X(6.5) = 23) = = (c) The conditional probability P(X(6.5) = 23 | N(6.5) = 30) = %. %. Note: For parts (b) and (c), enter your answers as percentages but do not type the percent sign.3. (20%) Let W1 < W2 < W3 < W4 < W5 < W6 be the order statistics of n = 6 independent observations from f(x) = x/2; 0 < x < 2. (a) Find E(W6)Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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