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

• B
• Schwann
In summary, a function that meets the criteria of a probability density function must be non-negative and have a total integral of 1. This means that even if the function reaches zero at certain points, it can still be a PDF. Additionally, such a function can be used to construct a probability distribution and there exists a probability space on which a random variable with this distribution exists. While a density function may not be unique, it is unique almost everywhere.
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?

Many pdf functions have zeros.

Schwann
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##.

Schwann
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).

Schwann
Schwann said:
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.

Last edited by a moderator:
Schwann and sysprog

## 1. Can a PDF value ever be equal to zero?

Yes, a PDF (probability density function) value can be equal to zero at certain points. This means that the probability of a random variable taking on a specific value at that point is zero.

## 2. What does it mean if a PDF value is equal to zero?

If a PDF value is equal to zero, it means that the random variable has no chance of taking on that specific value at that point. This could be because the value falls outside of the range of possible values for the random variable or because the probability of that value occurring is extremely low.

## 3. Are there any implications if a PDF value is equal to zero?

Yes, there are implications if a PDF value is equal to zero. It means that the probability of that specific value occurring is zero, and therefore it will not contribute to the overall distribution of the random variable. This could affect the accuracy of statistical analyses and predictions.

## 4. Can a PDF value be equal to zero for all points?

No, a PDF value cannot be equal to zero for all points. This would result in a constant function with a total probability of zero, which is not a valid probability distribution. A PDF must have at least one point where the value is greater than zero.

## 5. How can I interpret a PDF value of zero?

A PDF value of zero means that the probability of a random variable taking on a specific value at that point is zero. This could indicate that the value is an outlier or that it is not a possible value for the random variable. It is important to consider the context and range of possible values when interpreting a PDF value of zero.

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