Let X, Y be random variables, suppose X is N(0, 1) and Y = e2X. a) Obtain the density function of Y . b) Calculate E(Y) and V ar(Y).
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Let X, Y be random variables, suppose X is N(0, 1) and Y = e2X.
a) Obtain the density
b) Calculate E(Y) and V ar(Y).
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- 3. Let random variables X and Y be independent with joint density fe(x, y). Let Ix (0) and Iy (0) be the Fisher information of X and Y, respectively. Prove that the Fisher information I(x,y)(0) = Ix(0) + Iy(0).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.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.
- Suppose X is a random variable with Gaussian density with mean = 0 and standard deviation o = 1, this variable is transformed to the a.v. And through transformation Y=T(X)=2X2−1. Find: a) The rank of Y. b) The density function of Y.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 ∼ 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 random variables having joint density 4xy, 0sxs1,0sysl f(x,y) = 0, otherwise E(X) is equal to O A. 2/3 В. 4/9 C. 2/9 D. 4/3Suppose that the random variables X and Y have a joint density function f(x,y).prove that Cov(X,Y)=0 if E(X|Y=y) does not depend on yLet X and Y be independent random variables each having density function 2r fx (x) = 2 e * for x20 elsewhere Find: E (X+ 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?The input to a noisy communication channel is a binary random variable X with P (X = 0) = P (X = 1) = The output of channel Z is given by X+ Y, where Y is the addictive noise introduced by the channel. Assuming that X and Y are independent and Y = N (0,1), find the density function of Z.b) Let Y1, Y2.,Y5 be independent random variables with probability density function y 1 e 4 4 f(y)= ,y > 0 ,elsewhere 3 Determine the distribution and parameter of V = EY; using the method of moment i=1 generating function. Hence, find the mean of V using the moment generating function.
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