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![Q2. Here, the joint density function of two random variable X and Y are,
S(2,9) = {clE*y*
3y² + x²y³), 1<x < 4, 0<y< 4
0,
otherwise
(a) Find the value of c.
(b) Find P(2<x<5, 1<y<6)
(c) Find the marginal density function of x and y.
(d) Are x and y independent?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5faa7035-dfc3-479a-b113-31be2ca1585e%2Fca4e136a-5abd-499e-97cb-313eb0cdbe79%2Fl3whpe1_processed.png&w=3840&q=75)
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- Suppose you have two independent random variables X ~ Y ~ Exponential(A2), A1 > 0, d2 > 0. Find: Exponential(A1) and (a) The joint density function for (X,Y). (b) P( > 1). (c) P(X +Y > 1).2. Let X and Y be jointly continuous random variables with joint PDF x + cy2 0, OSXS1,0Sys1 elsewhere a) Find the constant c Find the marginal PDF's fy(x) and fy(y) c) Find P(OSXS1/2,0SYS1/2) b)If X, and X, are two random variables having joint density function 2. f(x,,x, ) = +8x,x,* ,0Sx, S 2,0s x,sI 8. Find i. P(X, I) P(X, <1/ X, <0.5) P(X,+ X, <1) ii. iii. iv.
- 5. Two random variables X and Y have a joint probability density function 0 < y < x < 2 fxy(x, y) = elsewhere (a) Find the marginal density functions of X and Y. (b) Are X and Y statistically independent?The continuous random variables X and Y are statistically independent and have marginal density functions fx(x) = 2x, 0 1. y2' Calculate the probability P(X < 0.5, Y < 2)Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 04. Let X and Y be independent, continuous random variables with densities and fx(x) = = fy (y) = = 0 {} if 0 < x < 2 otherwise. y if 027. If the joint density of the random variables X and Y is [emin{x,y) - 1] e-(x+y) if 0 < x, y <∞ f(x, y) = 0 otherwise, then what is the marginal density function of X, where nonzero?Let (X, Y) be a pair of jointly continuous random variables with joint p.d.f. 3 f (x, y) : =, 0sx< 1, 2²< y< 1. 2' (a) Find the marginal p.d.f. fx(x) of X and the marginal p.d.f. fy (y) of Y. (Ь) Find the covariance Cov(X, Y) of X and Y. (c) Find the conditional p.d.f. fy|x(y|x) of Y given X = x, for any x E (0, 1). (d) Find the conditional expectation E(Y|X = 1/2) of Y given X = 1/2. 2.Find the density of U = Y1 +Y2, where Y1 and Y2 are independent random variables with densities (yı – 1), 3 < Yı < 4, (y2 + 2), 0 < Y2 < 1, fy, (y1) : fy,(y2) = otherwise, otherwise.1. The joint density for the random variables (X, Y) is given below. for 0SS1,0SySI 0. for clsewhere (a) Find the marginal density of X (b) Find the marginal density of Y (c) Find VarlX] (d) Find VarV] (e) Are X and Y statistically independent? () Find P(X>0.6) (R) I EXY-0.8, find the correlation between X and YSuppose that the random variables Y1 and Y2 have joint probability density function, f(y1, y2), given 6y y2, 0syı < y2, y1 + y2 < 2 fV1, y2) = - 0, elsewhere Derive the marginal density of Y1. Derive the marginal density of Y2.(Hint: this will have to broken down into two parts!!!) Derive the conditional density of Y2 given Y1 = y1. Find P(Y2 < 1.1 | Y1 = .60). %DRecommended 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