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Is there a solution to the expected value/variance for a Gaussian with random phase:

$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$

?

For $t=0$, the solution is for example given here: Resultant probability distribution when taking the cosine of gaussian distributed variableResultant probability distribution when taking the cosine of gaussian distributed variable

Is there a solution to the expected value/variance for a Gaussian with random phase:

$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$

?

For $t=0$, the solution is for example given here: Resultant probability distribution when taking the cosine of gaussian distributed variable

Is there a solution to the expected value/variance for a Gaussian with random phase:

$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$

?

For $t=0$, the solution is for example given here: Resultant probability distribution when taking the cosine of gaussian distributed variable

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divB
divB
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expected value of cosine wirh Gaussian phase

Is there a solution to the expected value/variance for a Gaussian with random phase:

$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$

?

For $t=0$, the solution is for example given here: Resultant probability distribution when taking the cosine of gaussian distributed variable