5.4.2. For the previous problem, also assume X(t) is a Gaussian random process.
(a) Determine the name and parameters of the density of Z. Hint: Outputs of LTI
systems with Gaussian inputs are Gaussian. You determined the needed parameters
in the previous problem.
(b) Determine P(Z >1).
(c) Determine the name and parameters of the density of Z − W. Hint: Linear combinations of Gaussians are Gaussian. Note that E[Z − W] = E[Z] − E[W] and
Var(Z − W) = Var(Z) + Var(W) − 2Cov(Z, W).
(d) Determine P(Z−W <1).
![5.4.2. For the previous problem, also assume X(t) is a Gaussian random process. (a) Determine the name and parameters of the density of Z. Hint: Outputs of LTI systems with Gaussian inputs are Gaussian. You determined the needed parameters in the previous problem. (b) Determine P(Z >1). (c) Determine the name and parameters of the density of Z − W. Hint: Linear combinations of Gaussians are Gaussian. Note that E[Z − W] = E[Z] − E[W] and Var(Z − W) = Var(Z) + Var(W) − 2Cov(Z, W). (d) Determine P(Z−W <1).](https://i0.wp.com/electricalstudent.com/wp-content/uploads/2022/08/542.png?w=1170&ssl=1 "5.4.2. For the previous problem, also assume X(t) is a Gaussian random process. (a) Determine the name and parameters of the density of Z. Hint: Outputs of LTI systems with Gaussian inputs are Gaussian. You determined the needed parameters in the previous problem. (b) Determine P(Z >1). (c) Determine the name and parameters of the density of Z − W. Hint: Linear combinations of Gaussians are Gaussian. Note that E[Z − W] = E[Z] − E[W] and Var(Z − W) = Var(Z) + Var(W) − 2Cov(Z, W). (d) Determine P(Z−W <1).")
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