# Difference between Probability and Probability Density

This question is from DeGroot's "Probability and Statistics" :

Unbounded p.d.f.’s. Since a value of a p.d.f.(probability density function) is a probability density, rather than a probability, such a value can be larger than $1$. In fact, the values of the following p.d.f. are unbounded in the neighborhood of $x = 0$:$$f(x) = \begin{cases} \frac{2}{3}x^{-\frac{1}{3}} & \text{for 0<x<1,} \\ 0 & \text{otherwise.} \\ \end{cases}$$

Now, I don't know how the p.d.f. can take value larger than $1$.Please let me know the difference between the probability and probability density.

• – Henry Aug 21 '16 at 23:07

Simply put:

$\rho(x) \delta x$ is the probability of measuring $X$ in $[x,x+\delta x]$. With

$\rho(x):=$ probability density.

$\delta x:=$ interval length.

A probability will be obtained by computing the integral of $\rho(x)$ over a given interval (i.e. the probability of getting $X\in [a,b]$ is $\int_a^b \rho(x) dx$. While $\rho(x)$ can diverge, the integral itself will not, and this is due to the fact that we ask that $\int_\mathbb{R}\rho(x) dx=1$, which means that the probability of measuring any outcome is 1 (we are sure that we will observe something). If the integral over the whole range gives 1, the integral over a smaller portion will give less than 1, because p.d.f. can't be negative (a negative probability is meaningless).

Probability density is a "density" FUNCTION f(X). While probability is a specific value realized over the range of [0, 1]. The density determines what the probabilities will be over a given range. What does it mean to have a probability density?

The probability density function for a given value of random variable X represents the density of probability (probability per unit random variable) at that particular value of random variable X.

Now, I don't know how the p.d.f. can take value larger than 1

It is in this sense that probability density can take values larger than 1.

The specific values $f(x)$ of the density function $f$ are the probability densities, and they express "relative probabilities", and the main point is that for a (measurable) subset $A$ of possible values (now $A\subseteq\Bbb R$), we have $$\int_Af\ =\ P(X\in A)$$ if the random variable $X$ has distribution described by $f$. In particular, $\int_{\Bbb R}f=1$, though its specific values, as shown by the given unlimited example, can be greater than $1$.

• Will you please explain "relative probability" term? – Silent Oct 10 '13 at 16:26

If you have a continuous random variable X with a value between 0 and 3 and the probability (is always between 0 and 1) that X will occur between 2 and 2.1 is say 0.2, the probability density (probability rate) will be 0.2/0.1 = 2. when you multiply the probability density by the interval of the event (2*0.1 = 0.2), you will get the probability.

• You're answering a question that is 4½ years old, and have an accepted answer, you're probably not going to help anybody. Furthermore this looks more like an example than an actual description of the difference that was asked for. And you should learn MathJax/LaTeX and use it to format your posts. – Henrik May 13 '18 at 21:07

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