How well does a single data point fit a distribution? I have to come up with a way to measure the 'quality' of a distribution for a research project. 
We collect data over a a period of time $t_0$ through $t_1$ and then estimate the distribution that fits this data. At a later time, $t_2$, we collect a single piece of data. Is there a way to measure the likelihood of the initial estimated distribution being 'correct', given this new data point? 
We assume that the data points are independent random variables.
Thanks! 
 A: Call your original sample $X = \{x_1, \ldots, x_n \}$. If you have a density estimate $\hat{f}(x|X)$ computed from your data, then the likelihood of your new data point $u$ is just $\mathcal{L}(u)=\hat{f}(u)$. Higher values of $\mathcal{L}$ imply it's more plausible that the new data point was obtained from the same underlying distribution of the original sample.
A: Not without some additional assumptions.
Imagine the distribution the data are drawn from happens to consist of $n+1$ possible equally likely values, and that in $(t_0,t_1)$ you observe $n$ of those values. At time $t_2$ you observe the $(n+1)$th possible value for that distribution.
It doesn't matter how far away that last point is from the rest of the data, the observed sample is no more "weird" than it taking any of the other values for the last point; indeed the MLE of the distribution (the empirical cdf) is the closest possible to the true distribution.
A: Let your inferred distribution from the data collected in $(t_0, t_1)$ have parameters $\theta$.  The likelihood over $\theta$ is proportional to the density of the point observed at $t_2$ given $\theta$:
$$L(\theta \mid x_{t_2}) = f(x_{t_2} \mid \theta).$$
