Example 9-49. For the exponential distribution f(x) = e-x, x 2 0; show that the cumulative distribution function (c.d.f.) of X(n) in a random sample of size n is : F, (x) = (1 – e-x)n. Hence prove that as n → o, the c.d.f. of X(n) – log n tends to the limiting form exp [-{exp (– x)}] – ∞< x < o.
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![Example 9-49. For the exponential distribution f(x) = e-x, x 2 0; show that the
cumulative distribution function (c.d.f.) of X(n) in a random sample of size n is :
F, (x) = (1 – e*)n. Hence prove that as n → 0, the c.d.f. of X(n) – log n tends to the limiting
form exp [-{exp (– x)}] – ∞< x < o.
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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?3. A random variable X has a cumulative distribution function if 0 0.5). (e) Find P(X < 1.25). (f) Find P(X = 1.25).a) A university projects that enrollments are going to decline as the pool of college-aged applicants begins to shrink. They have estimated the number of applications for coming years to behave according to the function a = f(t) = 6,500 – 250t where a equals the number of applications for admission to the university and t equals time in years measured from this current year (t = 0). i. What class of function is this? ii. What is the expected number of applications 5 years from today? 10 years? iii. Do you think this function is accurate as a predictor indefinitely into the future? iv. What kinds of factor would influence the restricted domain on t?
- The 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.3. Let X1, and E- X; are independent. , Xn be iid with exponential distribution with mean 0. Show that (X1+X2)/ E=1 X;2. A random variable X has a p.d.f. f(x) given by √(1-x)6 02. Determine whether each of the following distributions are exponential family. For those that are, find a set of complete, minimum sufficient statistics. For those that are not, find a sufficient statistic. (a) Location Exponential: f(x) = e-(-0), x > 0. (b) f(x) = 0(1+x)(1+0), x > 0 (c) logN(u,02): f(x) = 1 a exp{-(log x-μ)²), x > 0. αν2πσ2 },1 1637. Prove or disprove that the following function is a cumulative distribution function (cdf). If the function is a cdf, is the random variable continuous or discrete? Why? (Note: This is a special case of a logistic distribution.) Fx(x)= 1 1+ e-z -∞<<∞.Example 17.15. Let X1, X2, ..., X, be a random sample from a distribution with p.d.f. : ••.. (x)- f(x, 8) = e¯*-®),0If X has the exponential distribution given by: f(x) = 4*(1 - x/2); 1 ≤ x ≤ 2. Find E(x).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)…SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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