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I may be measuring the length of a stick. I then want to see what is the probability of one measurement under a model.

When using a continuous model, the probability of a single number is zero:

$$ \mathbb P( \text{Length}=\text{length} \mid \text{Model}=\text{model} ) = 0 \>. $$

What is right way to calculate the probability the measurement? Should I calculate the probability of an interval? If the precision of the ruler is 1 mm, should I use the $\text{length} \pm 0.5 \text{mm}$ interval, i.e.,

$$ \mathbb P( \text{Length} - 0.5 \text{ mm} \leq \text{Length} \leq \text{length} + 0.5 \text{ mm} \mid \text{Model}=\text{model} ) \>? $$

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In a continuous model, probabilities of single values are indeed always zero. Instead, one use the density of the probability distribution, which is uniquely defined almost everywhere, thus at the observed measurement. This leads to the statistical concept of likelihood, which defines a quantitative assessment of a given sample in a continuous model.

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  • $\begingroup$ Also note that if you want to calculate the density function with the language you're using, it is the derivative of F(x), where F(x)=P(x<length|Model=model). $\endgroup$
    – David Robinson
    Jan 6, 2012 at 21:30
  • $\begingroup$ I'm trying to avoid the probability density. I may have also some discrete random variables. Is it a good practice to mix f(x) with P(Y=y) in a probabilistic model? $\endgroup$
    – Ivo Danihelka
    Jan 6, 2012 at 22:57

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