# Absolutely continuous r.v. vs. continuous r.v.

• AxiomOfChoice
In summary, the conversation discusses the difference between absolutely continuous random variables and continuous random variables. A random variable is called absolutely continuous if there exists a measurable function f≥0 such that P\{a<X<b\}=\int_a^b f(x)dx. If the cumulative distribution function (CDF) is continuous, the random variable is considered continuous. However, there is some disagreement among authors about the definition of continuous random variables. Additionally, there are singular random variables, such as the Dirac delta distribution, which are continuous but not absolutely continuous. These discussions also touch on the concept of distributions in probability and Shwartz distributions, which are not the same but have some overlap.
AxiomOfChoice
"Absolutely continuous r.v." vs. "continuous r.v."

I've recently come across the term "absolutely continuous random variable" in a book on measure theoretic probability. How am I supposed to distinguish between AC random variables and just continuous random variables?

Maybe if you consider X as a function (with random values) into some measure space,

then X is AC as a function?

A random variable X is called absolutely continuous if there exists a measurable function f≥0 such that

$$P\{a<X<b\}=\int_a^b f(x)dx.$$

If F is the cdf, that is, if $F(t)=P\{X\leq t\}$, then

$$F(t)=\int_{-\infty}^t f(x)dx$$

It can be checked that F is a continuous function.

Now, I think that the notion of continuous random variable depends on the author. Some define absolutely continuous and continuous as the same thing. Others say that X is continuous if the cdf F is continuous. In that case, we have seen that every absolutely continuous random variable is continuous. But there are (weird) continuous random variables that are not absolutely continuous. In practice, the interesting notion is clearly absolutely continuous, and not continuous.

I believe the Dirac delta distribution is an example of a distribution which is considered continuous but not absolutely continuous.

micromass said:
A random variable X is called absolutely continuous if there exists a measurable function f≥0 such that

$$P\{a<X<b\}=\int_a^b f(x)dx.$$

If F is the cdf, that is, if $F(t)=P\{X\leq t\}$, then

$$F(t)=\int_{-\infty}^t f(x)dx$$

It can be checked that F is a continuous function.

Now, I think that the notion of continuous random variable depends on the author. Some define absolutely continuous and continuous as the same thing. Others say that X is continuous if the cdf F is continuous. In that case, we have seen that every absolutely continuous random variable is continuous. But there are (weird) continuous random variables that are not absolutely continuous. In practice, the interesting notion is clearly absolutely continuous, and not continuous.

Right, but doesn't it come down to the same thing as f being AC as a function?

SW VandeCarr said:
I believe the Dirac delta distribution is an example of a distribution which is considered continuous but not absolutely continuous.
That's too loose.

Bacle2 said:
Right, but doesn't it come down to the same thing as f being AC as a function?
And that's too strict.
Another way to look at a continuous random variable is that such a random variable must have P(X=x) for all x. Yet another way to look at it is that the continuous random variable has a continuous CDF. A random variable with a Dirac delta distribution violates both.

A random variable is absolutely continuous if the CDF has a derivative, call it f(x), except over a space of measure zero. There's nothing saying this function f(x) has to be continuous.

A random variable that is continuous but not absolutely continuous is called a singular random variable. One example of such a random variable would be one whose CDF is everywhere continuous but nowhere differentiable. The CDF doesn't have to be nowhere differentiable to qualify as singular. It just has to be non-differentiable over a space with a non-zero measure.

D H said:
Another way to look at a continuous random variable is that such a random variable must have P(X=x) for all x. Yet another way to look at it is that the continuous random variable has a continuous CDF. A random variable with a Dirac delta distribution violates both.

'Distributions which are induced by some locally integrable function are said to be regular. Other distributions (such as the delta distribution) are said to be singular. (As an exercise, prove that the delta distribution is not induced by any locally integrable function).'

SW VandeCarr said:

'Distributions which are induced by some locally integrable function are said to be regular. Other distributions (such as the delta distribution) are said to be singular. (As an exercise, prove that the delta distribution is not induced by any locally integrable function).'

Distributions in probability are not the same thing as Shwartz distributions, aka generalised functions. There is overlap, but they are different spaces and have different properties.

pwsnafu said:
Distributions in probability are not the same thing as Shwartz distributions, aka generalised functions. There is overlap, but they are different spaces and have different properties.

Well, I won't disagree with you, but the limit of the Gaussian distribution as the variance approaches zero is the Dirac delta distribution.

http://math.stackexchange.com/quest...delta-function-and-delta-as-limit-of-gaussian

Last edited:

## What is the difference between an absolutely continuous random variable and a continuous random variable?

An absolutely continuous random variable is a type of continuous random variable that has a probability density function (PDF) that is defined for all possible values. This means that the probability of any specific value occurring is zero. In contrast, a continuous random variable has a PDF that is defined for all possible values, but the probability of any specific value occurring is not necessarily zero. Essentially, all absolutely continuous random variables are continuous, but not all continuous random variables are absolutely continuous.

## How are the properties of an absolutely continuous random variable different from those of a continuous random variable?

The properties of an absolutely continuous random variable and a continuous random variable are largely the same, as both are types of continuous random variables. However, the main difference is that an absolutely continuous random variable has a PDF that is differentiable, while a continuous random variable does not necessarily have a differentiable PDF. This means that the PDF of an absolutely continuous random variable can be used to calculate probabilities and expected values using integration, while a continuous random variable may require other methods.

## What types of distributions can be considered absolutely continuous random variables?

Any distribution that has a PDF that is defined for all possible values and is differentiable can be considered an absolutely continuous random variable. Some common examples include the normal distribution, uniform distribution, and exponential distribution.

## Are all absolutely continuous random variables unbounded?

No, not all absolutely continuous random variables are unbounded. While many distributions used as examples of absolutely continuous random variables, such as the normal distribution, are unbounded, there are also bounded distributions that can be considered absolutely continuous. For example, a uniform distribution on a finite interval is an absolutely continuous random variable, but it is bounded by the endpoints of the interval.

## Can an absolutely continuous random variable have discrete values?

No, an absolutely continuous random variable cannot have discrete values. As mentioned before, the probability of any specific value occurring for an absolutely continuous random variable is zero. This means that the values of an absolutely continuous random variable must be continuous and cannot be discrete.

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