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The oscillatory frequency is defined by $F=1/T$. Given that T has a normal distribution $N(µ_t,\sigma_t$) how do I calculate the frequency distribution based on period $µ_t,\sigma_t$ ?

I assume that $\sigma_f = 1/\sigma_t$ does is not true, and this should be evaluated by using the delta method.

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The distribution you are asking for is the reciprocal normal distribution, which is emphatically not another normal distribution, and in fact has no defined mean or variance. See mean and variance of reciprocal normal distribution which asks basically the same question.

The question you may mean to be asking is that if in the position domain the distribution is $N(µ_t,\sigma_t$) what is the frequency decomposition. This is a matter of doing a Fourier transform. Here the mean is and sigma are defined.

That is, the distribution you may be looking for is $$ \tilde{f}(k) = \int \frac{1}{\sigma_t\sqrt{2\pi}} e^{\frac{(t-\mu_t)^2}{2\sigma_t}} e^{-2\pi ikt} dt $$ with some factor of $2\pi$ to some power thrown in, depending on definitions.

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