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- 2)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the maximum likelihood estimator (MLE) of parameter θ.3. Let X, Y be continuous random variables with joint pdf f(x, y). Calculate the joint cumulative distribution function of (X, Y) for each of the pdfs below. (a) f(x, y) = 6e-3x-2y, x, y ≥ 0. (b) f(x, y) = x + ³y²,0 ≤ x, y ≤ 1.Find the moment-generating function for a random variable W with density w-1 e 1< w < 2, f(w) e 0, else.
- Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X3,..., X, has the following probability density function (0%e-0 f(x; 0) = x! 0, I for x = 0,1,2,... elsewhere Show that for any &> 0 and Sn = 1X₁, lim P(|S₁ - 0 ≥ ) = 0. 71-00 Show that a statistic S,, in a) is the maximum likelihood estimator of the parameter 8. Let 0₁ = X₁+2X2+2X₁-X4 and ₂ = (X₁ + X2 + X3 + X4) be two unbiased estimators of 0. Which one of the two estimators is more efficient? ) What is the Cramer-Rao lower bound for the variance of the unbiased estimator of the parameter 0? Use the one-parameter regular exponential family definition to find the functions, h(x), c(0), w(0) and t(x).Let X₁, X2, X3,..., Xn denote a random sample of size n from the population distributed with the following probability density function: e) g) of Ô, Suppose that Ô is a uniform minimum variance unbiased estimator of 0. Show that the variance (0+1)² Var (ê) = ! f(x; 0) = {(0 + 1)xº, 0, n if 0 < x < 1 elsewhere Use part d) of question 2 to show that is a consistent estimator of 0. Show that ₁ ln(X;) is a sufficient estimator of 0. [Hint: Use one-parameter regular exponential family]. Estimate using the method of moment estimator.a) Show for the constant c=g, the function f(x) = c(x+x²) for 0 < x < 1 is a proper density of a probability distribution on (0, 1). b) Find the corresponding cumulative distribution function F(x). c) Find the expected value and standard deviation of the random variable X with density
- Let X₁, X₂, X3,...,Xn be a random sample of n from population X distributed with the fol probability density function: 1 ze=20, f(x;0)=√√2n0 0, 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) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.b) Let X₁, X2, X3,...,Xn be a random sample of n from population X distributed with the following probability density function: f(x; 0) = 1 -20₁ V2πθ e 0, if - ∞0 < x < 0 otherwise (i) Find the parameter space of 0. (ii) Find the maximum likelihood estimator of 0. (iii) Check whether or not the estimator obtained in (ii) is unbiased. (iv) Find the Fisher information in this sample of size n about the parameter 0.Let Z₁ = X₁-μx ~N(0,1), and Wi σχ = Yi-HY~N(0,1), for i = 1,2,3,...,10, then: oy i) State, with parameter(s), the probability distribution of the statistic, T = ii) Find the mean and variance of the statistic T = Σt, wi? Σ, Z² Σt=1Zi 4i=1 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of such that P(T > B) = 0.01, where T = Σ₁₁Z₁² + 10₁ W₁². ẞ Σ{1Z2 Σi,Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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