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

1. Aug 21, 2012

### 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?

2. Aug 21, 2012

### Bacle2

Re: "Absolutely continuous r.v." vs. "continuous r.v."

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

then X is AC as a function?

3. Aug 21, 2012

### micromass

Re: "Absolutely continuous r.v." vs. "continuous r.v."

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.

4. Aug 21, 2012

### SW VandeCarr

Re: "Absolutely continuous r.v." vs. "continuous r.v."

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

5. Aug 21, 2012

### Bacle2

Re: "Absolutely continuous r.v." vs. "continuous r.v."

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

6. Aug 21, 2012

### D H

Staff Emeritus
Re: "Absolutely continuous r.v." vs. "continuous r.v."

That's too loose.

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.

7. Aug 21, 2012

### SW VandeCarr

Re: "Absolutely continuous r.v." vs. "continuous r.v."

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).'

8. Aug 21, 2012

### pwsnafu

Re: "Absolutely continuous r.v." vs. "continuous r.v."

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.

9. Aug 21, 2012

### SW VandeCarr

Re: "Absolutely continuous r.v." vs. "continuous r.v."

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: Aug 21, 2012