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![10. Let X and Y have the joint probability density function
for 0 < x <y<2
-{*²
f(x, y) =
elsewhere.
What is the marginal density function of X where it is nonzero?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe591e3e-c1e9-4180-ad99-642a1f427277%2F282df7bd-5c0f-42b5-b7d6-36d9592628ff%2Fnt8zcp7_processed.png&w=3840&q=75)
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- A random variable X has probability density function for 0 < x < 2, 4 f(x) = { +(2x –- 3) for 2 < x < 3, otherwise. Calculate µ, the mean value of X. Give your answer to three decimal places.Q1. Let x be a random variable with density function 2 (k+1)x 0 < x < 1 0 otherwise 9 = f(x): Find the moment generating function of x.Let X and Y have the following joint probability density function: fX,Y(x,y) = 1 / (x2 y2), for x>1, y>1 0, otherwise Let U = 5 X Y and V = 8 X. In all question parts below, give your answers to three decimal places (where appropriate). (a) The non-zero part of the joint probability density function of U and V is given by fU,V(u,v) = A vB uC for some constants A, B, C. Find the value of A.
- Let X and Y be two random variables with joint density function f(x,y) = (3 − x + 2y) / 60, for 1 < x < 3, 0 < y < 5. Find the marginal density functions. (Check that they are nonnegative and integrate to 1 before continuing further) Are X and Y independent? Explain.Let X1, X2, . X6 be an i.i.d. random sample where each X, is a continuous random variable with probability density function f(x) = e-(-0) , x > 0 Find the probability density function for X(6).The cumulative distribution function of the continuous uniform distribution between con- stants a and b is given by F(x) = P(X ≤ x) = x-a - a x b (a) The probability density function is f(x) = F(r). Find the form of f(x). (b) Find the derivative of f(x) for x = [a, b]. Is f(x) decreasing, increasing or flat in this region? (c) Does f(x) have a single maximum in the region [a,b]? If so, what is it, or if not, why not?
- 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.]Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0The lifetimes X and Y of two components in a machine have the joint density function f(x, y) = (1/4)*(2y + 2 − x) for 0 < x < 2, 0 < y < 1 and f(x, y) = 0 otherwise. (a) What is the probability density of the time until neither of two components is still working? (b) What is the probability distribution of the amount of time that the lifetime X survives the lifetime Y? KINDLY REQUEST YOU TO PROVIDE ME WITH COMPLETE SOLUTIONLet 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 is4. Let F(x) = S f(z)dz where f(x) is a valid probability density function. Find an expression for d F'(x) = F(x) in terms of the function f. Hint: Look up the Fundamental Theorem of Calculus on Wikipedia and read the corollary.25. Let X and Y be continuous random variables with joint density function for 0SEE MORE QUESTIONSRecommended 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