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- Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3, ..., Xn has the following probability density function 0xe-0 x! 0, a) b) c) f(x; 0) = for x = 0,1,2,... elsewhere 1 n Show that for any & > 0 and S₂ = ₁X₁, lim P(|S, − 0| ≥ ɛ) = 0. - Zi=1 n n→∞ Show that a statistic Sn in a) is the maximum likelihood estimator of the parameter 8. Let Ô₁ X₁+2X₂+2X3-X4 = 4 ¹ and Ô₂ = ²(X₁ + X₂ + X3 + X4) be two unbiased estimators of 8. Which one of the two estimators is more efficient?2. A random variable X has a probability density function (pdf) given by , 0 |X > ).Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3,..., X has the following probability density function (exe-e a) b) c) f(x; 0) = x! 0, " for x = 0,1,2,... elsewhere Show that for any &> 0 and S₂ = 1X₁, lim P(|S₂ - 0 ≥ ) = 0. 12-00 Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 0. Let 0₁ = X₁+²x₂+2x₁-X and Ô₂ = (X₁ + X₂ + X3 + X4) be two unbiased estimators of 8. Which 4 one of the two estimators is more efficient? d) What is the Cramer-Rao lower bound for the variance of the unbiased estimator of the parameter 9? e) Use the one-parameter regular exponential family definition to find the functions, h(x), c(0), w(0) and t(x).
- 1) Let X₁, X₂, X3, ... Xn be a sequence of independent and identically distributed with the following probability density function 1-1/(x-0² e f(x; 0)=√√2π if - ∞ < x <∞o otherwise 0, a) Is f(x; 0) a member of one-parameter regular exponential family? If yes, write down the functions, h(x), c(0), w(0) and t(x). b) Find the maximum likelihood estimator of the parameter 0. c) Find the variance of the maximum likelihood estimator of the parameter 8. d) Which one between X and (X₁ + X₂ − 2X3 + 2X4) statistics is more efficient?Let X₁, X₂, X3,..., Xn denote a random sample of size n from the population distributed with the following probability density function: f) f(x; 0): = {CO.. ((0+1)xº, if 0b) Let X₁, X2, X3.....Xn be a random sample of n from population X distributed with the following probability density function: ze zo, f(x;0)=√2m0 0, (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. if -∞b) Suppose that X₁ and X₂ have the joint probability density function defined as f(x₁, x₂) = (x1x²,0x₁51, 0 ≤ x₂ ≤1 elsewhere Find: i) the value of w that makes f(x₁, x₂) a probability density function. ii) the joint cumulative distribution function for X₁ and X₂. iii) P (X₂ ≤ | X₂ ≤ ³).b) Suppose that a sequence of mutually independent and identically distributed continuous random variables X₁, X₂, X3X₁ has the probability density function 1 (x-8)² 2 for -00Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3, ..., Xn has the following probability density function a) b) c) f(x; 0) = 0xe-0 x! 0, J for x = 0,1,2,... elsewhere Show that for any & > 0 and S₂ = = =₁ X₁, lim P(|S₂ − 0| ≥ ɛ) = 0. n n-00 one of the two estimators is more efficient? Show that a statistic S₁ in a) is the maximum likelihood estimator of the parameter 8. Let Ô₁ = X1+2X2+2X3-X4 4 ¹ and Ô₂ = ²(X₁ + X₂ + X3 + X4) be two unbiased estimators of 0. WhichThe time, X, for clearing side effects of water purification, in days, has the following cumulative distribution F: 1 F(x) = 0 for x 9 2 a) What is the probability density function for X for 1 5? c) What is the probability X 7? e) What is the probability that X > 6.3? f) What is the probability that X > 7 given the X > 6.3? g) Calculate the 70th percentile of X. h) What is the expected value of X? i) What is the expected value of x2 ? j) What is the variance of X? k) What is the probability that X is more than 0.1 below its expected value?1. Consider a random variable X with probability density function given as: 0 if x < 0, 1 if 0 ≤ x ≤ 1, f(x) 2 1 if 1 < <∞. -2x² Verify if f(x) is indeed a density function. If it is, find the distribution function of X.Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON