# Question: Suppose a random variable X has a CDF given by

Suppose a random variable X has a CDF given by Fx (x) and similarly, a random variable Y has a CDF, Fy (y) . Prove that the function F(x,y) = Fx (x) Fy (y) satisfies all the properties required of joint CDFs and hence will always be a valid joint CDF.

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Let and be zero- mean jointly Gaussian random variables with a correlation coefficient of and unequal variances of σ2X and σ2Y. (a) Find the joint characteristic function, Φ X, Y (ω1, ω2). (b) Using the joint ...A pair of discrete random variables has a PGF given by (a) Find the means, E [M] and E [N]. (b) Find the correlation, E [MN]. (c) Find the joint PMF, P M, N (m, n). Let and be jointly Gaussian random variables with E [X] = 1, E [Y] = –2, Var (X) = 4, Var (Y) = 9, and ρX, Y. Find the PDF of Z = 2X – 3Y – 5. Suppose and are independent, zero- mean Gaussian random variables with variances of σ2x and σ2y respectively. Find the joint PDF of Z = X2 + y2 and W = X2 – Y2 Repeat Exercise 5.66 Suppose In figure 5.7 and P i = 1/3, i = 1, 2, 3. Determine the mutual information for this channel. IfPost your question