. Find the joint density function of the random variables Z = X + Y and W = 2X −Y . What is the correlation coefficient between these two random variables
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Let X and Y be independent Gaussian random variables, each distributed according to (0, σ2).
1. Find the joint density
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- Let X be a uniform random variable over the interval1 to 4 and Y is exponential with a mean of 2. If the correlation between X and Y is 0.5 then compute the covariance between 3X and -5Y i.e. Cov(3X,-5Y)Let X1, X2, and.X3 be independent and normally distributed random variables with E(X1) 4, E(X2) = 3, E(X3) = 2, Var(X1) = 1, Var(X2) = 5, Var(X3) = 2. Let Y = 2X1 + X2 – 3X3. Find 2. the distribution of Y.Let the random variable X have the density function is given by сх, 0Let X and Y be two continuous random variables with joint probability density (3x function given by: f(x.y)%= 0sysxLet X = ((X1,Y1),...,(X, Yn)) be a sample from a bivariate normal population. Show that the distributions of X form a five-parameter exponential family and identify п, В, Г, and h.Let X and Y be independent Exp(2) random variables. Define W = X + Y. a) Determine the correlation between X and W. The joint pdf of X and W is: fx,w(x, w) = 1^(2) e^(-1w) I{OYearly returns on two assets (in percent) are denoted by rx and ry for asset X and Y respectively. Their joint distribution is multivariate normal with with mean (6, 2) and a covariance matrix 0.4 -0.2 -0.2 0.3 • Find the best linear approximation of the relationship between returns for a given return of asset X.Let x, x2, ..., Xn be a random sample from a Poisson distribution with parameter 2. Find ML estimator of 2. Also find its variance.Q. Let the joint pmf of x and y be defined by f (x,y) = (x+y)/32 where x = 1,2 and y = 1,2,3,4Suppose that X and Y are unknowns with E ( X ) = 10, Var ( X ) =4, E ( Y ) = 12 and Var ( Y ) = 100. In addition, suppose that the correlation coefficient of X with Y is .8. Then what is the expected value of XY?Let the random variable X have the density function is given by F(e) =*. Os s5, lo, elsewhere. 1) The value of the constant C=; 2) The expected value of X is E(X)=; 3) The expected value of X2 is E (x2) =; 4) The variance of X is given by Var (X) =o? =.Let X and Y are two random variables with p.d.f as F(x,y) = 3x^2y+3xy^2,0 < x< 1,0 < y < 1. (i) Find conditional density of X given Y. (ii) Find the mean, mode and standard deviation of r.v Y. (iii) Correlation between X and Y. (iv) Check that X and Y are independent or not?SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON