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So i know of many interpretations of the PDF of Random Variables.

  • How does the notion of pdf's relate to the volume in the bivariate Random Variables case? Is the volume under the curve an indication of how much probability mass is located at a particular section?

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  • I find it to be quite confusing when we one hand when we are using the idea of length times density to give us the area which happens to be the probabilistic mass.

  • Intuitively, is mass and volume & area the same thing? When we have images like these... I struggle to see the connection between mass and density as i feel it is easier to see the connection between density and volume under curve.

  • OR is it rather the idea that for example the area under a curve for (1-dimensional RV) is the PROBABILISITIC MASS, the mass which is born of the LENGTH times DENSITY? (which i am guessing the units of density is always:

mass divide (area or length) depending on the dimensions of the RVs.

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I do not know if I understand all your questions correctly. But for the first question you are right, the mass of the pgf at a particular area can be interpreted as the probability of that area.

The interpretation 3D is similar to the interpretation in 2D. In 2D, indeed, you can take a small length $[x,x+dx]$, and integrate the pgf over that interval. This results in an the area which is $P(x\leq X\leq x+dx)$.

In 3D however, we have two random variables, lets say $X$ and $Y$. Now, you cannot just take an interval $[x,x+dx]$, since the pgf tells you about the probabilities of $X$ and $Y$ together. Therefore, you take an area $[x,x+dx],[y,y+dy]$. Then, again you integrate the pgf over that region. However, since the region now is 2D, this integral gives you a volume instead of an area. This volume can be interpreted as $P(x\leq X\leq x+dx,y\leq Y\leq y+dy)$.

Therefore in 2D, the probabilistic mass can be seen as an area, and in 3D as a volume. However, you could of course also take higher dimensional variables (relating for example $X,Y$ and $Z$). Then the interpretation as area/volume becomes more difficult, and therefore the notion of probabilistic mass is used.

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