Let XaZ + b where Z~ N(0, 1) and a and b are real numbers with a > 0. Find the density function of X and hence show that X~ N(μ=b, o² = a²).
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- A speedboat has two engines, A and B. The joint density function of the lifetimes of the two engines, measured in hours, is: f(A, B) = .125 (A + B) %3D defined on the square: 0 < x < 2 and 0< y < 2. Calculate the probability that the speedboat fails during its first 84 minutes of operation, according to: The boat fails upon either engine failing: The boat fails only when both engines have failed:Q3 The joint density function of X and Y is given by: Find: 1- P(X> 1nY0,y>0 otherwise fxy(x, y) = {Ce STRIJThe density function is often used as a model for the lengths of life of physical systems. Suppose Y has the Weibull density just given. Find: a) the density function of U = Ym b) E(Yk) for any positive integer k
- a) Find the conditional density of X, given Y = y, and that of Y , given X = x. Make sure to define the support of each density function. b) Find the density function of Z = XY . Make sure to define the support of f (z).kindly make it as detailed as possible.Let A > 0. Suppose (X, Y) has joint probability density function given, for all x, y E R, by 3. fx,y(x, y) = { Ay exp(-y(x+ A)), if a 2 0 and y > 0 0, otherwise. (i) Calculate the marginal density of Y. (ii) Let y > 0. Find the the conditional probability density function of X given Y = y. Interpret your answer. (iii) Calculate E[X|Y] and prove that E[X] = +oo. Hint: You can use without a proof that -dy = +oo. %3D
- Verify the claim that p (r) = p (x, y) dy, where p(r, y) and p (r) denote %3D the joint density and marginal density, respectively.The joint probability density function for X and Y is f(x, y). Find P(X < 0.7| Y = 0.5). Remember, you need to find the conditional density for X at Y = 0.5, then do the appropriate integration. f(x, y) = { i (2x + y) 0 < x < 1, 0Suppose that (X,Y)' has a density function given by for a > 1, y > 0, f(x, y) = 10, otherwise. Determine the distribution of X?Y.Let X denote how long it takes (in hours) to do homework set #2 in MAT271E, and assume that the density function of X is given by х - 2 if 2Please answer i,ii, and show steps if possible. Do not randomly write wrong answers if don't know how to solve.compute the mass along the x-axis, the moment (M) around x=0, and the center of mass given the data in exercise 6. Show full and complete procedure. the letter "ρ" represents the density functiion. meaning ρ is the density function shown. Solve for exercise 3Recommended 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