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$$\psi(x,t)=\frac{1}{(2\pi)^{1/4}\sqrt{\alpha+i\beta t}}e^{i(k_0x-\omega_0 t)}e^{\frac{-(x-v_g t)^2}{4(\alpha+i\beta t)}}$$

$$|\psi(x_s,t)|^2=\frac{1}{\sqrt{2\pi(\alpha^2+\beta^2t^2)}}\exp\left(\frac{-\alpha(x_s-v_gt)^2}{2(\alpha^2+\beta^2t^2)}\right)$$

$$|\psi(x_s,t)|^2=\frac{\pi}{\sigma_x^2}\exp\left(-\frac{(x_s-ct)^2}{2\sigma_x^2}\right)$$

$$\psi(x,t)=\left(\frac{\alpha}{2\pi}\right)^{1/4}\frac{1}{\sqrt{\alpha+i\beta t}}e^{i(k_0x-\omega_0 t)}e^{\frac{-(x-v_g t)^2}{4(\alpha+i\beta t)}}$$

$$|\psi(x_s,t)|^2=\sqrt{\frac{\alpha}{2\pi(\alpha^2+\beta^2t^2)}}\exp\left(\frac{-\alpha(x_s-v_gt)^2}{2(\alpha^2+\beta^2t^2)}\right)$$

$$|\psi(x_s,t)|^2=\frac{1}{\sqrt{2\pi}\sigma_x}\exp\left(-\frac{(x_s-ct)^2}{2\sigma_x^2}\right)$$

So if we start with our Gaussian momentum wavefunction $\psi(k)=e^{-\alpha(k-k_0)^2}$, where $\alpha=\frac{\hbar^2}{2\sigma_p^2}=\sigma_x^2$, we can follow this procedure to find the position wavefunction as a function of time:

$$\psi(x,t)=\sqrt{\frac{\pi}{\alpha+i\beta t}}e^{i(k_0x-\omega_0 t)}e^{\frac{-(x-v_g t)^2}{4(\alpha+i\beta t)}}$$

$$|\psi(x_s,t)|^2=\frac{\pi}{\sqrt{\alpha^2+\beta^2t^2}}\exp\left(\frac{-\alpha(x_s-v_gt)^2}{2(\alpha^2+\beta^2t^2)}\right)$$

So if we start with our Gaussian momentum wavefunction $\psi(k)=e^{-\alpha(k-k_0)^2}$, where $\alpha=\frac{\hbar^2}{2\sigma_p^2}=\sigma_x^2$, we can follow this procedure to find the position wavefunction as a function of time (and then normalize said wavefunction, because the authors of that source apparently didn't bother to do so):

$$\psi(x,t)=\frac{1}{(2\pi)^{1/4}\sqrt{\alpha+i\beta t}}e^{i(k_0x-\omega_0 t)}e^{\frac{-(x-v_g t)^2}{4(\alpha+i\beta t)}}$$

$$|\psi(x_s,t)|^2=\frac{1}{\sqrt{2\pi(\alpha^2+\beta^2t^2)}}\exp\left(\frac{-\alpha(x_s-v_gt)^2}{2(\alpha^2+\beta^2t^2)}\right)$$

Now, we discretize our continuous time parameter. Our detector interacts with the environment, but those interactions are not instantaneous: every interaction that would allow a detection has some associated timescale $\Delta t$ (for example, detectors based on ionization would have a timescale associated with the amount of time an incoming particle takes to ionize an atom). So we can model our detector as a device that periodically "checks" to see whether it interacted with a particle. So now we have a set of discrete times, $t=0, \Delta t,2\Delta t,...$ at which the metaphorical dice are rolled. But this time, each time these metaphorical dice are rolled, the probability is different. For conciseness, let's label $P(n\Delta t)$ as $P_n$ for the following section.

Now, we discretize our continuous time parameter. Our detector interacts with the environment, but those interactions are not instantaneous: every interaction that would allow a detection has some associated timescale $\Delta t$ (for example, detectors based on ionization would have a timescale associated with the amount of time an incoming particle takes to ionize an atom). So we can model our detector as a device that periodically "checks" to see whether it interacted with a particle. So now we have a set of discrete time intervals, $t=0, \Delta t,2\Delta t,...$ during which the metaphorical dice are rolled. 

But this time, each time these metaphorical dice are rolled, the probability is different. And it's clear that we can't actually use the probability at a particular instant, either, because that would imply that we know what the "phase" of the detector's interactions are, which we don't. So instead, we average the probability over one interaction timescale. Let $P_n$ be the probability that a detector detects a particle in the interaction timescale interval $(n\Delta t, (n+1)\Delta t)$:

$$P_n=\frac{1}{\Delta t}\int_{n\Delta t}^{(n+1)\Delta t} P(t)dt$$