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![othen Let z= x² + xy var (2) = vas (8²4xy) var(x2) + Varesy) + 2 Cor (x2xy) = E649-{6x2737 Ebr>y) - (E(xy)]²+ a[ E(+²y) - E()](http://img.homeworklib.com/questions/0995e750-6988-11eb-8452-6db1e1624cc4.png?x-oss-process=image/resize,w_560)
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X & Y jointly gaussian zero mean Groom waste common variance 6² & correlation coefficient pto...
Let ˜x and ˜y be zero-mean, unit variance Gaussian random variables with correlation coefficients, . Suppose...
Let ˜x and ˜y be zero-mean, unit variance Gaussian random
variables with correlation coefficients, . Suppose we form two new
random variables using linear transformations:
Let and be zero-mean, unit variance Gaussian random variables with correlation coefficients, p. Suppose we form two new random variables using linear transformations: Find constraints on the constants a, b, e, and d such that ù and o are inde- pendent.
6.72 Let Y =X+N where X and N are independent Gaussian random variables with different variance and N is zero mean. (a) Plot the correlation coefficient between the “observed signal” Y and the “desire...
6.72 Let Y =X+N where X and N are independent Gaussian random variables with different variance and N is zero mean. (a) Plot the correlation coefficient between the “observed signal” Y and the “desired signal” X as a funtion of the signal-to-noise ratio (b) Find the minimum mean square error estimator for X given Y (c)Find the MAP and ML estimators for X given Y (d) Compare the mean square error of the estimators in parts a, b, and c.
2. (30 Points) X and Y ~ N (0,4) are two jointly Gaussian random variables, and...
2. (30 Points) X and Y ~ N (0,4) are two jointly Gaussian random variables, and E(XY) = 3 a. (10 Points) Find their joint PDF, f (x,y). b. (10 Points) Find the mean and variance of Z = X +Y. c. (10 Points) Find the mean and variance of Z = X + Y + 2.
Let X be Gaussian with zero mean and unit variance. Let Y = |X|. Find: a)...
Let X be Gaussian with zero mean and unit variance. Let Y = |X|. Find: a) The PDF fY (y) b) The mean E[Y ] c) Here X is uniform in (0, 1), but now you are asked to find a functiong(·) such that the PDF of Y = g(X) is ?2y 0≤y<1fY (y) = 0 otherwise
Let X and Y be two independent Gaussian random variables with common variance σ2. The mean of X i...
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...
3 -0.751 (X1,X2, X3) be jointly Gaussian with ux (1,-2,3) and Cx 1. Let X =...
3 -0.751 (X1,X2, X3) be jointly Gaussian with ux (1,-2,3) and Cx 1. Let X = 3 0.25 4 L-0.75 0.25 Hint: If a set of random variables (RVs) are jointly Gaussian, then any subset of those RVs are also jointly Gaussian. Similarly, adding constants to (or taking linear combinations of) jointly Gaussian RVs results in jointly Gaussian RVs. Using this property you can solve problem 1 without using integration. When appropriate, you may express your answer by saying that...
X is a Gaussian random variable with zero mean and variance ơ2 This random variable 5...
X is a Gaussian random variable with zero mean and variance ơ2 This random variable 5 20 points is passed through a quantizer device whose input-output relation is g(z) = Zn, for an x < an+1, 1 N where In lies in the interval [an, Qn+1) and the sequence fa, a2, al z-00, aN41 # oo, and for i > j we have ai > aj. Find the PMF of the output random variable Y g(X). aN+1) satisfies the conditions
Suppose that X is a Gaussian Random Variable with zero mean and unit variance. Let Y=aX3...
Suppose that X is a Gaussian Random Variable with zero mean and unit variance. Let Y=aX3 + b, a > 0 Determine and plot the PDF of Y
1. The random variable X is Gaussian with mean 3 and variance 4; that is X...
1. The random variable X is Gaussian with mean 3 and variance 4; that is X ~ N(3,4). $x() = veze sve [5] (a) Find P(-1 < X < 5), the probability that X is between -1 and 5 (inclusive). Write your answer in terms of the 0 () function. [5] (b) Find P(X2 – 3 < 6). Write your answer in terms of the 0 () function. [5] (c) We know from class that the random variable Y =...
3, X and Y are two jointly Ga a. b. th G (μ, μ., σ. σ2y, py). ussian random variables WI What is the "most likely" value of X given Y? If Z = X+Y, find the correlation coefficient between...
3, X and Y are two jointly Ga a. b. th G (μ, μ., σ. σ2y, py). ussian random variables WI What is the "most likely" value of X given Y? If Z = X+Y, find the correlation coefficient between Z and Y assuming for this part that the means of X and Y are zeros.
3, X and Y are two jointly Ga a. b. th G (μ, μ., σ. σ2y, py). ussian random variables WI What is the...
Let ˜x and ˜y be zero-mean, unit variance Gaussian random
variables with correlation coefficients, . Suppose we form two new
random variables using linear transformations:
Let and be zero-mean, unit variance Gaussian random variables with correlation coefficients, p. Suppose we form two new random variables using linear transformations: Find constraints on the constants a, b, e, and d such that ù and o are inde- pendent.
2. (30 Points) X and Y ~ N (0,4) are two jointly Gaussian random variables, and E(XY) = 3 a. (10 Points) Find their joint PDF, f (x,y). b. (10 Points) Find the mean and variance of Z = X +Y. c. (10 Points) Find the mean and variance of Z = X + Y + 2.
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...
3 -0.751 (X1,X2, X3) be jointly Gaussian with ux (1,-2,3) and Cx 1. Let X = 3 0.25 4 L-0.75 0.25 Hint: If a set of random variables (RVs) are jointly Gaussian, then any subset of those RVs are also jointly Gaussian. Similarly, adding constants to (or taking linear combinations of) jointly Gaussian RVs results in jointly Gaussian RVs. Using this property you can solve problem 1 without using integration. When appropriate, you may express your answer by saying that...
X is a Gaussian random variable with zero mean and variance ơ2 This random variable 5 20 points is passed through a quantizer device whose input-output relation is g(z) = Zn, for an x < an+1, 1 N where In lies in the interval [an, Qn+1) and the sequence fa, a2, al z-00, aN41 # oo, and for i > j we have ai > aj. Find the PMF of the output random variable Y g(X). aN+1) satisfies the conditions
1. The random variable X is Gaussian with mean 3 and variance 4; that is X ~ N(3,4). $x() = veze sve [5] (a) Find P(-1 < X < 5), the probability that X is between -1 and 5 (inclusive). Write your answer in terms of the 0 () function. [5] (b) Find P(X2 – 3 < 6). Write your answer in terms of the 0 () function. [5] (c) We know from class that the random variable Y =...
3, X and Y are two jointly Ga a. b. th G (μ, μ., σ. σ2y, py). ussian random variables WI What is the "most likely" value of X given Y? If Z = X+Y, find the correlation coefficient between Z and Y assuming for this part that the means of X and Y are zeros.
3, X and Y are two jointly Ga a. b. th G (μ, μ., σ. σ2y, py). ussian random variables WI What is the...
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