Consider a random sample X1,X2 ···Xn from CDF F(x) = 1 − 1/x for x ∈ [1,∞) and zero otherwise.
Find the limiting distribution of n ln X1:n.
Consider a random sample X1,X2 ···Xn from CDF F(x) = 1 − 1/x for x ∈...
2. Consider a random sample XI, X2 otherwise. Xn fronn CDF F(x) = 1-1/z for z e [ X) = 1-1/1 for x 1, oo) and zero (a) Find the limiting distribution of X1:n, the smallest order statistic. (b) Find the limiting distribution of X1: (c) Find the limiting distribution of n In X1m
Let X1, X2, .. , Xn be a random sample of size n from a geometric distribution with pmf =0.75 . 0.25z-1, f(x) X-1.2.3. ) Let Zn 3 n n-2ућ. Find Mz, (t), the mgf of Žn. Then find the limiting mgf limn→oo MZm (t). What is the limiting distribution of Z,'? Let X1, X2, .. , Xn be a random sample of size n from a geometric distribution with pmf =0.75 . 0.25z-1, f(x) X-1.2.3. ) Let Zn 3...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
Consider a random sample X1, X2, ..., Xn is from f(x) = e-(x-a) , x > a > 0. A) Find the method of moments estimator for a B) Find the MLE for a
6. Let X1,..., Xn be a random sample from Uniform (0, 1). a) Find the exact distribution of U = – log(X(1)) where X(1) = min(X1, X2,..., Xn). b) Find the limiting distribution of n(1 – X(n)), where X(n) = max(X1, X2, ..., Xn).
Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = (1/θ)e^(−x/θ) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ. Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = θe^(−θx) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ.
2. Consider a random sample Xi,Xy otherwise. Xu frorn CDF F(x) 1-1/x for x e [1,00) and ze (a) Find the limiting distribution of Xim, the smallest order statistic. (b) Find the limiting distribution of XT (c) Find the limiting distribution of n In X1m
Let X1, X2, ...,Xn denote a random sample of size n from a Pareto distribution. X(1) = min(X1, X2, ..., Xn) has the cumulative distribution function given by: αη 1 - ( r> B X F(x) = . x <B 0 Show that X(1) is a consistent estimator of ß.
Assume that X1, X2, . . . , Xn denote a random sample from a population with the following probability density function : fX(x|α) = αβ / (α + βx)^2 , x > 0 where α > 0 and β > 0. find the limiting distribution of nβX(1).
4. Let X1, X2, ..., Xn be a random sample from a distribution with the probability density function f(x; θ) = (1/2)e-11-01, o < x < oo,-oo < θ < oo. Find the NILE θ.