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Normal distributionWhat is the intuition behind the probability density function of a continuous random variable? Integrating it within two points provides the probability that is associated between two points, but if you plug a single value into $f(X=x)$, it outputs a value. What is the intuitive understanding of this value for a given x?

For example, in the above image, the value of the red curve at x=0 is 0.4. How can 0.4 be interpreted?

statBeginner
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    See this thread: http://stats.stackexchange.com/questions/133369/the-total-area-underneath-a-probability-density-function-is-1-relative-to-wh , there is at least few nice definitions and multiple examples. – Tim Jan 29 '15 at 07:53
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    The notation $f(X=x)$ is inappropriate for a pdf $f$: $f(x)$ is the value of the density at the real value or vector $x$, not a pseudo-probability that $X$ is equal to $x$. – Xi'an Jan 29 '15 at 07:55

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It is a density value. Density only has meaning when it is collectively considered over an area or volume.

mcvz
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The value of a continuous density function at a given point is also called its likelihood. Yes, the probability of getting any specific number in the range is exactly equal to 0—there are uncountably many options. However, it remains true that observations may cluster, and some values are still more likely than others.

For example, the probability that a standard normal variable $Z$ takes on the value $0$ is exactly the same that it takes on the value $1\times 10^8$, namely 0. But you will see $0$s much more often. This is reflected in the value of the pdf at the point, as any interval containing 0 is going to have a larger area, and thus larger probability, than any equal-length interval not containing 0 (the normal is symmetric and unimodal at 0).

So, informally speaking, the value of 0.4 is not useful for its magnitude in and of itself (although it is very useful in maximum and relative likelihood analyis), but it can be compared with other values on the curve to see which values may be more likely. But, as said earlier, probability only has positive value for intervals of continuous distributions.

Avraham
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In the case of a discrete function it may output a value. A continuous function will output zero for a particular point since it represents an infinitesimal. Here is a good article explaining it from ground up:

The idea of a probability density function

Hope that helps.

Emir
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  • Here is what i mean. In the question, i added an image from wikipedia. The value of the function at x=0 is 0.4. However, the value at any specific point should be negligible. So what does 0.4 correspond to? – statBeginner Jan 29 '15 at 08:23
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    As @mcvz said it is a density: the probability per unit of $x$ or the probability of falling between $x$ and $x+1$ if the probablity density remains constant over that interval (and it does not, so be careful with that interpretation). – Maarten Buis Jan 29 '15 at 08:36
  • It's the output of the probability function. E.g. p(x). However, for continuous distributions you can't just use that value. You have to integrate. In the discrete case integrating or not for a point value give you the same result. In the continuous case it does not. – Emir Jan 29 '15 at 08:44