Here X1, X2,....,X2 a random indepedent variable which has the same distribution and the same law of X. Sn = 1/nΣ(Xi) a. we have ε>0. what is Lim (n-> +∞) P(|Sn| >= ε) ?
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Here X1, X2,....,X2 a random indepedent variable which has the same distribution and the same law of X.
Sn = 1/nΣ(Xi)
a. we have ε>0. what is Lim (n-> +∞) P(|Sn| >= ε) ?
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- Q3. (a) Define a function g(x) of a random [ 1; if x2X, 0; otherwise variables by g(x)% = Where x, is a real number -o< X, <∞ Show that E [g (X)] = 1-F, (x,)25. Let X₁,..., Xn be a sample from a population with o² = Var(X) <∞, μ = E(X) and let T = X² be an estimate of μ². (a) When µ ‡ 0, find the asymptotic distribution of √√n(T-μ²) using the delta method. μ = (b) When μ 0, find the asymptotic distriution of nT using P(nT ≤ t) = P(-√t ≤ √nX ≤ √t). Compare your answer to the answer in part (a). (c) Find the limiting laws of √n(Ỹ − µ)² and n(Ñ — µ)².1. 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).2(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 0Let X1,..., Xn be independent and identically distributed random variables with Var(X1) < ∞. Show that n 1 EjX;→p EX1. п(п + 1) j= Note. A simple way to solve a problem of showing Yn →p a is to establish limn EYn = a and limn Var(Yn) = 0.3. 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 0SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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