2) Two statistically-independent random variables, (X,Y), each have marginal probability density,...
Can you show the steps clearly? Thank you 1) We construct a 2x1 Gaussian random vector via the canonical representation, -1 :4A1/2W+ m, where D , and Wi and W2 m- are statistically independent, zero-mean, unit-variance Gaussian. a) Find the mean and the covariance of X b) Let Y = cy , where c = ofY 4 Find the explicit formula for the probability density 1) We construct a 2x1 Gaussian random vector via the canonical representation, -1 :4A1/2W+ m,...
a) Let X and Y be two random variables with known joint PDF Ir(x, y). Define two new random variables through the transformations W=- Determine the joint pdf fz(, w) of the random variables Z and W in terms of the joint pdf ar (r,y) b) Assume that the random variables X and Y are jointly Gaussian, both are zero mean, both have the same variance ơ2 , and additionally are statistically independent. Use this information to obtain the joint...
1. (20 points) Let X (Xi, X, Xs) be a real random vector, where X, are identically dis- tributed and independent (ii.d.) zero-mean Gaussian real random variables. Consider the random vector Y given by where A is a 3 x 3 real matrix and b is a 3 x 1 real vector. Justify all your answers. (a) Find the covariance matrix Cx of x. (b) Find the mean vector EY] of Y (c) Express the covariance matrix Cy of Y...
Let X and Y be two independent Gaussian random variables with common variance σ2. The mean of X is m and Y is a zero-mean random variable. We define random variable V as V- VX2 +Y2. Show that: 0 <0 Where er cos "du is called the modified Bessel function of the first kind and zero order. The distribution of V is known as the Ricean distribution. Show that, in the special case of m 0, the Ricean distribution simplifies...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z max (X. Y) as the larger of the two, Derive the C.DF. and density function for Z. 2. Define W min(X,Y) as the smaller of the two. Derive the C.D.F.and density function for W 3. Derive the joint density of the pair (W. Z). Specify where the density if positive and where it takes a zero value....
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z-max (X, Y) as the larger of the two. Derive the C.D.F. and density function for Z. 2. Define Wmin (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W 3. Derive the joint density of the pair (W, Z). Specify where the density if positive and where it takes a zero...
7. Let X and Y be independent Gaussian random variables with identical densities N(0,1). Compute the conditional density of the random variable of X given that the sum Z = X + Y is known (i.e., XIX + Y)
9-A-Two random variables X and Y are independent and have marginal Probability Density Functions (PDF) shown below. Derive the PDF of the random variable Z = X+Y. Give an expression for the desired PDF and sketch it. 0.5e ; x>0 J1/2; 0<y<2 Jx0 ") 0 : f ) = 0 0; y<0
Let X and Y denote independent random variables with respective probability density functions, f(x) = 2x, 0<x<1 (zero otherwise), and g(y) = 3y2, 0<y<1 (zero otherwise). Let U = min(X,Y), and V = max(X,Y). Find the joint pdf of U and V.
1 Expectation, Co-variance and Independence [25pts] Suppose X, Y and Z are three different random variables. Let X obeys Bernouli Distribution. The probability disbribution function is 0.5 x=1 0.5 x=-1 Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y are independent. Meanwhile, let Z = XY. N(0,1). X and Y (a) What is the Expectation (mean value) of X? 3pts (b) Are Y and Z independent? (Just clarify, do not need to prove) [2pts c)...