In this problem, you will demonstrate that the Gaussian PDF in Equation ( 9.64) is in fact
Question:
methods. Define
time- varying characteristic function of the random process X (t).
(a) Starting from the diffusion Equation (9.63), show that the characteristic function must satisfy
Also, determine the appropriate initial condition for this differential equation. That is, find Φ (ω, t).
(b) Solve the first- order differential equation in part (a) and show that the characteristic function is of the form
(c) From the characteristic function, find the resulting PDF given by Equation (9.62).
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Related Book For
Probability and Random Processes With Applications to Signal Processing and Communications
ISBN: 978-0123869814
2nd edition
Authors: Scott Miller, Donald Childers
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