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

## Main Question or Discussion Point

"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?

## Answers and Replies

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Bacle2

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.

Bacle2

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?

D H
Staff Emeritus

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

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.

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.

See 1.24 re delta distribution. There might be some disagreement about this.

'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).'

pwsnafu