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- 7. Let X and Y be continuous random variables with joint probability density function given by fxy(xy): Sc, (0, 0 < x < y < 1 otherwise a) Find the value of c. b) Find the marginal distributions of X and Y. c) Find the conditional distribution of Y given X. d) Find the E(Y) and Var(Y).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: f(x;0)=√√2n0 0, -20₁ 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 X₁, X2, X3,..., Xn denote a random sample of size n from the population distributed with the following probability density function: I e) f) g) where k> 0 is a constant. f(x; 2) = 2-kxk-1e (k-1)! if x > 0 elsewhere " Justify whether or not  is a uniform minimum variance unbiased estimator of 1. Check whether or not the MLE of λ is a consistent. Suggest with a Fisher's factorization theorem, the sufficient estimator of 1.b) 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.3. If the probability density function of random variable is given by f(x)=1/sechx 1 ≤ x ≤ 2 a) Find the mean b) Find the total areaThe random variable X has the geometric distribution with probability mass function (pmf) Py(x) = q*p. x = 0,1, 2, ... 0 x). |A7 The random variable X has the binomial distribution with probability mass function () p*(1 - p)²-*, x = 0, 1,2; 0b) 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.Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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