2. Let X and Y be independent continuous random variables with the following density functions: 1 0Y)?/
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![2. Let X and Y be independent continuous random variables with the following density functions:
1 0< x < 1
f (x) =
0 otherwise
8y 0<y<0.5
f(y) =
otherwise
What is P(X >Y)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0d90012-896b-4f52-8f0b-e38034636c35%2F599224cd-943d-42bb-91de-c6cc03f16e9b%2Flqiymwn_processed.png&w=3840&q=75)
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- 8)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 joint probability density function of the order statistics X(2), X(4), X(6) for n = 7.7. Let X and Y denote two continuous random variables. Let f(x,y) denote the joint probability density function and fx(x) and fy (y) the marginal probability density functions for X and Y, respectively. Finally let Z = aX + bY, where a and b are non-zero real numbers. (e) Derive an expression for Cov(Z) as a function of Var (X), Var(Y) and Cov(X,Y). [You may use standard results relating to variance and covariance without proof, but these should be clearly stated.]2. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 0Let X and Y continuous random variable with joint density function f(x, y) = (²³/1 (2 - x - y) x - y) for 0 < x, y < 2; 0 < x + y<2 otherwise. 0 What is the conditional probability P(X < 1| Y < 1)?2. The continuous random variables X and Y have known joint probability density function f(x,y) given by £xy (x, y) = {(x + y) [(x+y)/8 0 ≤x≤ 2, 0≤y≤ 2 fxx otherwise Define the random variable z as Z = min(X,Y). a) Determine the probability density function fz(z) of the random variable z. b) Determine the probability Pr{0.5b) Let X and Y be jointly continuous random variables with the following joint density function (1/32)xy ,0 2) v. Evaluate P(Y > X – 2)Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0Let X and Y be continuous random variables with joint p.d. f. f(x,y) = {* (12x for 0< y< 2x < 1 elsewhere The conditional density function of Y given X = x is3. Let X and Y be two independent random variables with identical probability density function given by e- for x > 0 f(x) = elsewhere. What is the probability density function of W= max{X, Y } ?Q5. (10 marks) Suppose that the joint probability density function for X and Y is given by + 2y), 0 < x< 2,0 < y <1 f(x,y) = 4 0, Find the expected value, E(XY). %3D elsewhere1. Let X be a random number from (0, 1). Find the probability density function of Y = 1/X.4. Let X, Y be jointly continuous with density f(x, y) = = c(x - y)e-2 File Preview (a) Find the value of c. (b) Find the conditional distribution of X|Y. 0 < x<∞, -x < y < x otherwiseRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. FreemanMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman