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Consider a random process given by x(t) = A cos (2π fC t + θ), where A is a Rayleigh distributed random variable and θ is uniformly distributed in [0, 2π]. A and θ are independent. For any time t, the probability density function (PDF) of x(t) is:

Consider a random process given by x(t) = A cos (2π fC t + θ), where A is a Rayleigh distributed random variable and θ is uniformly distributed in [0, 2π]. A and θ are independent. For any time t, the probability density function (PDF) of x(t) is: Correct Answer Rayleigh

Since θ is uniformly distributed, so the PDF of cos(2πfc t + θ) will also be uniform.

Also, A and θ are independent.  So A and cos(2πfc t + θ) will also be independent.

As such,

PDF of x(t) = (PDF of A) × (PDF of Cos(2πfc t + θ))

But since PDF of cos(2πfc t + θ) is uniform., the PDF of x(t) will be PDF of A which is Rayleigh distributed random variable.

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