- Let Xq, Xzx m--Xh be the poisson d stributon with poanele d Find the minimum Lariance unbia sed estimetor a randem Sample frem
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- 2. Suppose alpha particle emissions of carbon-14 are counted by a Geiger counter each second. The emissions follow a Poisson process with the mean of 16 particles per second. Let W equal the time in seconds before the seventh particle is counted. a) Give the distribution of W. b) Find P(W ≤ 0.5).J.UZ! b) Let X₁, X₂,..., X and Y₁, Y₂,, Ym be random samples from populations with moment generating 25 functions Mx, (t) = e3t+t² and My, (t) = , respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ - X3 + x4 + X5. ii) What is the value of the sample size n, if P[Σ(X₁-X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-µy ≥10) < 0.04?7
- * 6) A triangular distribution has p.d.f.) f(x)= k(1- Sketch the p.d.f., and evaluate the constant k. (xl) for 1A kurtosis value near zero indicates which of the following? mesokurtic distribution O leptokurtic O platykurtic None of theseA.2) The time between successive customers coming to the market is assumed to have Exponential distribution with parameter lambda. a) If X1, X2. ..., Xp are the times, in minutes, between Successive customers selected randomly, estimate the parameter of the distribution. b) The randomly selected 15 times between successive customers are found as 1.8, 1.2, 0.8, 1.4, 1.2, 0.9, 0.6, 1.2, 1.2, 0.8, 1.5, 1.8, 0.9, 1.5 and 0.6 mins. Estimate the mean time between successive customers, and write down the distribution function. c) In order to estimate the distribution parameter with 0.4 error and 4% risk, find the minimum sample size.a) Suppose that X₁, X₂, X3,...,X, is a random sample of n from population X which has a c square distribution with one degrees of freedom. (1) Find lim Mg(t). 72-004. The arcsin distribution has the CDE and inverse CDF below. Using, .25, .74, and 46 as random numbers, generate the corresponding iterates from the arcsin distribution in radians. (Provide your answer in the form YO, Y1, Y2 and round answers to two decimal places.) p-¹(u) = (sin(u) ². F(x) = 2 πT arcsin (√),4. Let X,,X2, .. ,X„ be a random sample form a uniform distribution, f(X), = 0*(1– 0)*-X; 0 < x < 1. Assume T, = X, and T, = (HEx. Check the unbiasedness property for T, and T2. Find the MSE of T, and T2. Can you preferer one of them? Discuss your reasons. a. b. с.2.0.3 Problem 3: Central Limit Theorem Here we will verify the Central Limit Theorem and reproduce plots similar to those from Wikipedia (https://en.wikipedia.org/wiki/Central_limit_theorem#/media/File:Dice_sum_central_limit_theorer a) Write a function that returns n integer random numbers, uniformly distributed between 1 and 6, inclusively. This represents n throws of a fair 6-sided die. The value that comes up at each throw will be called the "score". b) Generate a distribution of 1000 dice throws and plot it as a histogram normalized to unit area. Compute the mean ₁ and standard deviation o₁ of this distribution. Compare your numerical result to the analytical calculation. c) Generate 1000 sets of throws of N = 2, 3, 4, 5, 10, 20, 30 dice, computing the total sum of dice scores for each set. For each value of N, plot the distribution of total scores, and compute the mean and standard deviation on of each distribution. This should be similar to the plot at the link above. d) Plot the…b) Let X₁, X₂,..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25 functions Mx₁(t) = ³t+t² and My(t) = (₁-¹)²5, respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5. ii) What is the value of the sample size n, if P[Σ1(X¡ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - µy| ≥10) < 0.04?20. Let X1, X2, ables, and set X, be independent, Exp(a)-distributed random vari- Y1 = X(1) and Y = X(k) – X(k-1); for 2SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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