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Wikipedia says:

The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

and it also says.

Unlike a probability, a probability density function can take on values greater than one; for example, the uniform distribution on the interval $[0, \frac{1}{2}]$ has probability density $f(x) = 2$ for $0 ≤ x ≤ \frac{1}{2}$ and $f(x) = 0$ elsewhere.

How are these two things compatible?

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It can take a value greater than $1$, but only over a region with measure less than $1$. –  Qiaochu Yuan Feb 3 '12 at 22:22
Compute the integral in your example. Is it 1? –  GEdgar Feb 3 '12 at 22:33
Think of it from a physics perspective: the probability density function is like the density of some compressible fluid, whereas it's integral is like the mass. If you take a fixed mass of liquid and compress certain parts of it, you can make the local density as high as you like without changing the total mass. –  Nick Alger Feb 4 '12 at 0:07
Yeah, I feel a bit silly now. –  zenna Feb 4 '12 at 1:05

2 Answers 2

up vote 7 down vote accepted

Consider the uniform distribution on the interval from $0$ to $1/2$. The value of the density is $2$ on that interval, and $0$ elsewhere. The area under the graph is the area of a rectangle. The length of the base is $1/2$, and the height is $2$ $$ \int\text{density} = \text{area of rectangle} = \text{base} \cdot\text{height} = \frac 12\cdot 2 = 1. $$

More generally, if the density has a large value over a small region, then the probability is comparable to the value times the size of the region. (I say "comparable to" rather than "equal to" because the value my not be the same at all points in the region.) The probability within the region must not exceed $1$. A large number---much larger than $1$---multiplied by a small number (the size of the region) can be less than $1$ if the latter number is small enough.

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Remember that the 'pd' in pdf stands for "probability density", not probability. Density means probability per unit value of the random variable. That can easily exceed 1. What has to be true is that the integral of this density function taken with respect to this value must be exactly 1.

if we know a pdf function(e.g. normal distribution), and want to know the "probability" of a given value, say x=1, what will people usually do? To find the probability that the output of a random event is within some range, you integrate the pdf over this range.

Also see Why mvnpdf give probablity larger than 1?

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