Let (X, Y) be a uniformly distributed random point on the quadrilateral D with vertices (0,0), (2,0), (1,1) and (0,1). (a) Find the joint density function of (X; Y ) and the marginal density functions of X and Y . (b) Find E[X] and E[Y ]. (c) Are X and Y independent?
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Let (X, Y) be a uniformly distributed random point on the quadrilateral D with vertices (0,0), (2,0), (1,1) and (0,1).
(a) Find the joint density function of (X; Y ) and the marginal density
X and Y .
(b) Find E[X] and E[Y ].
(c) Are X and Y independent?
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- Q1) Discrete joint variables X and Y with probability density f(x,y) (pdf) are given in this table. Find: 1) The Covariance Cov(X,Y)? (Cov(X,Y) = MxY-MxMY) 2) The correlation (pxy) between X and Y where Pxy = Cov(X,Y) PXPY Y 3 fx(x) f(x,y) 1 2 1 1/4 1/4 0 X 2 0 1/4 1/4 fy(y) Note that 2 n=2 n=3 Px² = Σn²±²x² f(x, y) - μ and py² = Σ3y² f(x, y) – µ Zk=0Let (X, Y)' be a random vector with the following density function: (8xy if0sxsysl elsewhere Find the marginal density functions of X and Y. Are X and Y independent? f(x)=Let X ∼ Unif(0, 1). Let g(x) = e x and Y = g(X). (i) Find the density fX of the random variable X. (ii) Find the cdf FY (y) = P(Y ≤ y) of Y . (iii) Find the density fY of the random variable Y .
- Let X and Y be two continuous random variables with joint probability density function: 6e-(2x+3y)+k x21, y > 0 fxy (x, y) = otherwise a) Find the coefficient k. b) Find P(X 2). c) Find the marginal probability density functions of X and Y ( fx(x), f;(y)). d) Are X and Y independent?Let X and Y denote the amplitude of noise signals at two antennas. The random vector (X, Y) has the joint pdf – f(x, y) = k(x + y) 03. Let f be the density function of a random variable X, and let Y = (Y1, Y2, Y3) be a continuous random vector with density function %3! fY (y1, V2, ys) 6 f(v)f(v)/() 1{vn < ya < ya}. (a) Find the conditional density fy, Y,Y, (V, Vsly2) and argue that (Y I. Y) | Y. (b) If X has a uniform distribution on the unit interval (0, 1], deduce that (Y, Y) | (Y = V) has a certain bivariate uniform distribution. Over which set?1)A random vector z = (x y) T has joint probability density given by:a)Sketch the graph of the probability density function pxy(X, Y ).b)Determine the value of the constant C.c)Determine the mean mz vector.2 Let X, Y be independent and identically distributed with Uniform(0, 1) density. (a) Find the density of X − Y (b) Find the density of X/Y .1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.Let (X,Y) have the joint density function f(x,y) = { (6/5)(1+x+y) for 0 < x, y < 1, x+y<1 otherwise } (g) Find the correlation coefficient pxy (h) Find guv (u,v) if U = (X+Y) and V=X (i) Find Var[U]i) Find the joint probability density function of X1 and X2. ii) Using the transformation method, find the joint probability density function of Y1 = X21 and Y2 = X1 + X2 iii) Determine the marginal density function of Y2The joint density of X and Y is given by, ху, Iw(x, y) = (x² +: 0 l) b) Find the marginal probability distributions of X and Y. c) Find the conditional probability density function of Y given X=0.5 and calculate P(YSEE 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