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The probability of an event, $P(A)$, is a definite number.

I can espress $P(A)$ as $P(A)=p(a)da$ where $p(a)$ is the probability density function.

However, $da$ is NOT a definite value, so how it's possible to get a definite quantity calculating $p(a)da$ and ascertain that effectively $P(A)=p(a)da$ ?

Qwerto
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    Who says you can express $P(A)$ as $p(a)da$? –  Apr 24 '18 at 07:05
  • It is not a problem that is exclusive to probabilities. I suppose you have encountered integrals before and used the same notation. Plus, as Bungo noted, the link between the PDF and the actual probability is not the oone you wrote down. You could have a look at this post. – Bill O'Haran Apr 24 '18 at 07:08
  • So, how can i calculate the probability of a single event A with the probability density function ? And why is worth using the integral calculus ? – Qwerto Apr 24 '18 at 07:10
  • Is there a random variable somewhere in the picture? Where is this probability density function coming from? Some context would be helpful. –  Apr 24 '18 at 07:14
  • So, can't i use the probability density function for NON random events ? Why ? – Qwerto Apr 24 '18 at 07:15
  • How do you define your probability density function? –  Apr 24 '18 at 07:15
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    "The Probability density function is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable’s PDF over that range. $\Pr [a \leq X \leq b] = \int {a}^{b}f{X}(x),dx$." – Mauro ALLEGRANZA Apr 24 '18 at 07:16
  • Then, using the probability density function i lose the way to find the probability of a single value ? If yes, what is the advantage ? – Qwerto Apr 24 '18 at 07:21
  • If a random variable has a probability density function (which happens if and only if the random variable is absolutely continuous), then the probability that it assumes any single value is zero. –  Apr 24 '18 at 07:26
  • I have now understood that the probability that it assumes any single value is zero because a real number requires a infinite precision. But, why things work so well with integral calculus ? – Qwerto Apr 24 '18 at 07:31
  • You can compute probabilities using integral calculus by accepting various integration rules and that is fine. As to why integral calculus and probability theory are related is the subject of measure theory. You will need to do a course in that. – Paul Apr 24 '18 at 08:25

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