# Question

Two random variables are jointly Gaussian with means of μx = 2, μy = –3, variances of σ2x = 1, σ2y = 4, and a covariance of Cov (X, Y) = –1.

(a) Write the form of the joint PDF of these jointly Gaussian random variables.

(b) Find the marginal PDFs, fX (x) and fY (y).

(c) Find Pr (X < 0) and (Pr 9Y > 0) and write both in terms of Q- functions.

(a) Write the form of the joint PDF of these jointly Gaussian random variables.

(b) Find the marginal PDFs, fX (x) and fY (y).

(c) Find Pr (X < 0) and (Pr 9Y > 0) and write both in terms of Q- functions.

## Answer to relevant Questions

Two random variables have a joint Gaussian PDF given by (a) Identify σ2x, σ2y, and ρX, Y. (b) Find the marginal PDFs, f X (x) and f Y (y). (c) Find the conditional PDFs, f X| Y (x| y) and f Y| X (y| x) A pair of random variables has a joint characteristic function given by (a) Find E [X] and E [Y] (b) Find E [XY] and Cov (X, Y). (c) Find E [X2Y2] and E [XY3]. Let and be independent and both exponentially distributed with Find the PDF of Z = X –Y. Let and be independent zero- mean, unit- variance Gaussian random variables. Consider forming the new random variable U, V according to U = [X] cos(θ) –[Y ] sin(θ) V = [X] sin (θ – [Y] cos (θ). Suppose Z = X + jY is a circular Gaussian random variable whose PDF is described by Equation (5.70), Find the characteristic function associated with this complex Gaussian random variable, ΦZ (ω) = E [exp (jωZ)]. Do you ...Post your question

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