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 μ and standard deviation σ.Hence, the probability density function of f ( x ) should be 1 σ 2 π e − ( ( f ( x ) − μ ) / σ ) 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.

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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   μ and standard deviation   σ.Hence, the probability density function of   f  (  x  ) should be      1          σ              2        π                  e          −                        (          (          f          (          x          )          −          μ          )                      /                    σ                      )            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.

laure6237ma · Accepted Answer

Your proposed density function is not correct. Rather, the probability density function of the error   ε is  w  ↦      1          σ              2        π                  e                            −          (          (          w          −          μ          )                      /                    σ                      )            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  )  +  ε  .You will never have an exact match since the probability distribution of the error has no discrete component---it&#039;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   μ and   σ.

I have a prediction f ( x ) of some continuous process variable, based on an input va

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