# What's the probability of a match between a prediction and a measurement?

I have a prediction $f(x)$ of some continuous process variable, based on an input variable $x$ (think: location). The prediction is incorrect, with the error being normal distributed with expected value $\mu$ and standard deviation $\sigma$.

Hence, the probability density function of $f(x)$ should be

$$\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{((f(x)-\mu)/\sigma)^2}{2}}$$

Is this correct? (No it is not, see answer below)

Now, I have a measurement $m$ of the process variable, based on an unknown input variable $x_m$. $m$ is assumed to be correct, but quantized to integral numbers.

Given a set of $x_i$ together with their predictions $f(x_i)$, how can I compute a probability that $x_m$ was in the vicinity of $x_i$?

I apologize if the question makes no sense. Comments that help me improve the question are appreciated!

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Your proposed density function is not correct. Rather, the probability density function of the error $\varepsilon$ is $$w\mapsto\frac{1}{\sigma\sqrt{2\pi}} e^{\frac{-((w-\mu)/\sigma)^2}{2}}.$$ If the prediction is $f(x)$, and the true value of the thing to be predicted is $g(x)$, then $$f(x) = g(x)+\varepsilon.$$ You will never have an exact match since the probability distribution of the error has no discrete component---it's a continuous distribution. So the probability of an exact match is $0$. But you can speak of the probability that the prediction differs by no more than a specified amount from the true value. That probability would depend on $\mu$ and $\sigma$.