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Edited for clarity.

Let $X$ be a continuous random variable with a known probability distribution. Let $\mathbf{v}$ be a vector in $\mathbb{R}^n$.

What is the probability that $\mathbf{v}$ is a sample of $X$?

Michael Hardy
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Asger Jon Vistisen
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    Zero, because there are an infinite number of rv's $v$ can come from. – JMP Aug 02 '20 at 17:46
  • The problem relates to training a neural network to produce samples belonging to some particular random variable. So when the network yields a sample, how "good" is it at belonging to the actual random variable? That is required for the loss function during training. – Asger Jon Vistisen Aug 02 '20 at 17:56
  • @JMP : In infinite number of WHAT?? We are speaking here of just one random variable, not an infinite number of them. It is asserted to have a continuous distribution. That implies the probability that it is equal to a specified number is $0.$ However, the poster said $v$ is a "sample of real numbers", and I wonder what that means. In particular, I wonder if the poster meant a set of real numbers. Could the question mean "What is the probability that $X$ is a member of the set $v$?"? If, for example, $v$ were the interval from $2$ to $3,$ then that probability could be positive. – Michael Hardy Aug 02 '20 at 18:01
  • Normally, for good reasons, one learns the theory of probability before learning the theory behind "fitting", so it seems a bit odd to say that that is the "reverse" of something that is "typical". You throw a die; What is the probability that you get a $1\text{?}$ That seems to be what you're calling "reverse fitting." $\qquad$ – Michael Hardy Aug 02 '20 at 18:03
  • At this, point, it seems some work will be needed before I can be sure what it is you're asking. – Michael Hardy Aug 02 '20 at 18:04
  • Try 'given $n$ rv's, and sample came from one, which one?' – JMP Aug 02 '20 at 18:06
  • @JMP : ok, Now that makes some sense. – Michael Hardy Aug 02 '20 at 18:07
  • @JMP : But you said it has a known distribution. How can it have a known distribution when the question asks for the probability that the distribution was one of a family of distributions? – Michael Hardy Aug 02 '20 at 18:09
  • You said "given $n$ random variables". Did you intend these to have different probability distributions? – Michael Hardy Aug 02 '20 at 18:11
  • OK @MichaelHardy I have tried to clarify the question with an edit. Any help would be much appreciated. – Asger Jon Vistisen Aug 02 '20 at 18:16
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    ok, You said in a comment "given $n$ random variables". There's nothing about that in the question as it now stands. Does your comment about "$n$ random variables" still apply? – Michael Hardy Aug 02 '20 at 18:19
  • @MichaelHardy No, "n random variables" does not. – Asger Jon Vistisen Aug 02 '20 at 19:02
  • At this point I can still only guess what you're asking. Here's another guess: You have $n$ independent copies of $X,$ i.e. $n$ independent random variables all of which have the same known probability distribution. Call them $X_1,\ldots,X_n.$ And you're asking for $\Pr( (X_1,\ldots,X_n) = \mathbf v). \qquad$ – Michael Hardy Aug 02 '20 at 20:35
  • @MichaelHardy Correct. If the X stochastic vector consisted of independent random variables then it would be: $f_{X,Y} = f_X(x) \cdot f_Y(x)$, but now that they are identical, I can't see how to compute it. – Asger Jon Vistisen Aug 02 '20 at 23:49

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