High School Can PDF values be equal to zero at some given points?

[Schwann]

Suppose we have a function which looks like this:

It seems like it meets criteria of probability density functions: this function is asymptotic to zero as x approaches infinity and also it is not negative. My question is: if at some points this function reaches zero (as I have shown above), does that mean that in cannot be PDF?

 

[Dale]

Many pdf functions have zeros.

 

[mathman]

The only conditions for a function $f(x)$ to be a pdf are $f(x)\ge 0$ for all $x$ and $\int_{-\infty}^\infty f(x)dx=1$.

 

[member 587159]

An example of a well-known absolute continuous distribution with density function that is zero on a set of infinite measure is the gamma-distribution and the exponential distribution (which is a special case of the former).

 

[member 587159]

Suppose we have a function which looks like this:
View attachment 252880
It seems like it meets criteria of probability density functions: this function is asymptotic to zero as x approaches infinity and also it is not negative. My question is: if at some points this function reaches zero (as I have shown above), does that mean that in cannot be PDF?

If you have a Borel-measurable map $f:\mathbb{R}\to [0,\infty[$ such that $\int_\mathbb{R} f =1$, then we get a measure

$$\mu(A) =\int_A f d\lambda, A \in \mathcal{B}(\mathbb{R})$$

and this is a probability distribution. One can even show that there is a probability space on which there exists a random variable with this distribution. More formally, there exists a probability space $(\Omega, \mathcal{F},\mathbb{P})$ and a random variable $X: (\Omega, \mathcal{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ such that $\mu=\mathbb{P}_X$.

So to anwer your question: such a function is certainly a density function of some random variable on some probability space.

Last remark: a density function $f$ need not be unique, but it is unique almost everywhere.