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- 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 consistencyLet Xi be iid Exp(λ) random variables.1. Let Mn = max1≤i≤n Xi. Compute the CDF FMn (x).2. Using the identity P(X > x) = exp(−λx), show that if an = log n, and Yn = Mn − an, thenFYn(x) → e^(-e^-x).3. * What does this say, heuristically, about the order of magnitude of Yn?Each 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)…
- A single observation of a random variable having a hypergeometric distribution with N = 7 and n = 2 is used to test the null hypothesis k = 2 against the alternative hypothesis k = 4. If the null hypothesis is rejected if and only if the value of the random variable is 2, find the power of the test.If X1, X2, and X3 constitute a random sample of sizen = 3 from a Bernoulli population, show that Y =X1 + 2X2 + X3 is not a sufficient estimator of θ. (Hint:Consider special values of X1, X2, and X3.)8.- Given that X is a hypergeometric random variable with N = 10, n = 5, and r = 7: a. Display the probability distribution for X in tabular form. b. Compute the mean and variance of X. c. Graph p(x) and locate μ and the interval μ ±2σ on the graph. d. What is the probability that x will fall within the interval μ ±2σ ?
- A researcher is using a two-tailed hypothesis test with α = 0.01 to evaluate the effect of a treatment. If theboundaries for the critical region are t = ± 2.845, then how many individuals are in the sample?A. n = 23B. n = 22C. n = 21D. n = 20E. cannot be determined from the information given1. Consider a random variable that is normally distributed where = 60; = 2, and x= 55. Find the corresponding value of z and interpret its computed value.2. Each of 12 refrigerators of a certain type has been returned to a distributor because of the presence of a high-pitched oscillating noise when the refrigerator is running. Suppose that four of these 12 have defective compressors and the other eight have less serious problems. If they are examined in random order, let ? be the number among the first six examined that have a defective compressor. Compute (a) ?(? = 1), (b) ?(? ≥ 4), and (c) ?(1 ≤ ? ≤ 3).