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- M:14) Using Ordinary Generating Functions prove in how many ways the sum of 5 dices give n as a result?Everyday, James reverses his car from his driveway on to the road in such a way that there is a very small probability P that his car will be involved in a collision. ( i) Show that the probability that there will be no collision in a five-day week is (1−P)^5 and state one assumption that is made in your answer. (iii) Let P = 0.001. Check that the Poisson approximation can be used, and find the approximate probability that James will avoid a collision in 500 days.Let X1, X2, ... , Xn be a random sample from N(μ, σ2). Find the Moment Generating Function of X̅. If n = 16 and σ = 2, compute P(-1 ≤ X̅ - μ ≤ 1).
- T2 #4 Apr 5 Of the international passengers arriving at an airport, 1.5% are selected for luggage inspection. Let X be the number of passengers from a random sample of 400 who are sent for luggage inspection. Use Poisson approximation to X to find the probability that between 6 and 12 passengers inclusive, are sent for luggage inspection.Theorem: Suppose (Xn)n≥1 is a sequence of random variables with corresponding momentgenerating functions M_Xn , and X is a random variable with moment generating functionM_X such that for some δ > 0 we have M_X (t) < ∞ for all t ∈ (−δ, δ). If lim n→∞ MXn (t) = MX (t) for all t, then lim n→∞ F_Xn (x) = F_X (x) for all x where F_X is continuous. That is, if the moment generating functions of X_n converge to the moment generating function of X, then the distribution of X_n converges to the distribution of X. Use this to show that if Sn ∼ Binomial(n, λ/n ), then the distribution of Sn converges to Poisson(λ) as n → ∞.Give a recursive definition (with initial condition(s)) of (an) where (n = 1, 2, 3, . . . )
- Consider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θI got MLE = n/-∑logXi and UMVUE = (n-1)/∑logXi. Need help in proving consistencyShow that the following series is divergentEach of 14 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 9 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 6 examined that have a defective compressor. (I have figured out part "a" but need help with "b" and P(X ≤ 3) in "c") (a) Calculate P(X = 4) and P(X ≤ 4). (Round your answers to four decimal places.) P(X = 4) = P(X ≤ 4) = (b) Determine the probability that X exceeds its mean value by more than 1 standard deviation. (Round your answer to four decimal places.) (c) Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If X is the number among 25 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately)…
- Based on the information above, are the assumptions met? Based on the information above, are the assumptions met? No, because there is a simple random sample but the population is not normally distributed Yes, because there is a simple random sample and the sample size is large enough for the Central Limit Theorem Yes, because there is a simple random sample Yes, because there is a simple random sample and the population is normally distributed36. Over the last three years, a portfolio manager investing in large cap stocks had an average return of 25% when the small cap benchmark he selected returned 19.5%. Is the performance consistent with efficient market hypothesis? a. No, it is not. It is a violation of weak form EMH. b. No, it is not. It is a violation of semi-strong form EMH. c. No, it is not. It is a violation of strong form EMH. d. No, it is not. It is a violation of all forms of EMH. e. Yes, it could be. The manager selected an inappropriate benchmark.Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?