Understanding Probability Density Functions for Beginners

[member 428835]

hello all!

i am wondering about probability density functions. i know the area under a pdf gives the probability of an event, but i am having a difficult time seeing this. specifically, given a pdf we have $\int_a^b f(x) dx$ as the probability of an occurrence from $[a,b]$. what are the units of $f(x)$? why exactly is $f \times dx$ the probability, rather than just $f$ itself?

please illustrate with a histogram if you can. thanks!

 

[member 428835]

i should add, in the text I'm using we are given that, if $H(c, \Delta c, N)$ where $\Delta c$ is the slot width, $N$ is the number of realizations of the random variable. evidently $$B(c) := \lim_ {\substack{\Delta c \to 0 \\ N \to \infty}}\frac{H(c, \Delta c, N)}{\Delta c}$$

where $B(c)$ is a pdf. can someone help explain this? nothing was really said about $c$ or $H$ other than $H$ is the histogram. i assume $c$ is a random variable?

 

[Stephen Tashi]

*

what are the units of $f(x)$? why exactly is $f \times dx$ the probability, rather than just $f$ itself?

Think of "density" in terms of physical density - for example, let f(x) be the mass density of a rod given in terms of kilograms per meter at the point x. To find the mass of a rod between two points, you'd do an integration. The value of f(x_0) at the point x_0 is some value of density, not a value indicating mass.

You could say that the "units" of a pdf are "probability per unit length" (or "per unit area" etc.) , but I don't know if anyone has ever worked out a good way for abbreviating all the information that goes along with "probability" in the same system we use for units in physics. To speak of "the probability" of an event unambiguously, you have to define a "sample space" and an algebra of sets of events and a measure defined on that algebra. If we say that a particular formula "is a pdf" then we convey a lot of mathematical conventions with that short phrase. As far as I know, in physics "probability" is a "dimensionless" quantity. From that point of view, the "units" of a pdf are 1 over the unit of measure used on the sample space. .

 

[Stephen Tashi]

we are given that, if $H(c, \Delta c, N)$

You didn't say what H is. You only defined its arguments.

 

[member 428835]

thanks for the reply! yea, tho author of the text on states that $H$ is a histogram. nothing more is stated that i haven't already listed...it's pretty annoying.