Dirac delta distribution

In summary: But the probability is still infinite. You can think of it as the probability of getting some result in an experiment that measures the value of a random variable that has a very small variance. In summary, the Dirac delta function is not a true function but rather a distribution. It can be seen as the limit of a sequence of functions, such as a normalized Gaussian with infinitesimal variance. The integral of the Dirac delta function is unity, meaning that it has unit area under the curve. The Dirac delta function is often used in probability theory, representing the probability of getting a result in an experiment with a random variable that has a very small variance.
  • #1
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Previously I posted a question on the Dirac delta function and was informed it was not a true function, but rather a distribution. However, I have to admit I still did not understand why its integral (neg inf to pos inf) is unity. I've thought about this and came up with the following:

Consider a normal distribution with a variance of 0. Since the integral of a normal distribution is constrained to be unity, the mean must go to infinity as the variance approaches zero. This is the Dirac delta. Is this reasonable or am I wandering in the wilderness?
 
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  • #2
you're confusing probability distribution and distribution.

http://en.wikipedia.org/wiki/Distribution_(mathematics [Broken])

they're probably related cause a lot of the words in that article are in probability articles but not like you think they are.
 
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  • #3
Consider a normal distribution with a variance of 0. Since the integral of a normal distribution is constrained to be unity, the mean must go to infinity as the variance approaches zero. This is the Dirac delta. Is this reasonable or am I wandering in the wilderness?
The mean doesn't have to go to infinity. It can stay at zero. In that case the density function has as its "limit" the Dirac delta function at 0. In other words the limit of the probability distribution is a Schwartz distribution.
 
  • #4
mathman said:
The mean doesn't have to go to infinity. It can stay at zero. In that case the density function has as its "limit" the Dirac delta function at 0. In other words the limit of the probability distribution is a Schwartz distribution.

I don't know much about general distribution theory, but I read that Schwartz "legitimized" the Dirac delta function by redefining it as a functional. In any case, I'm happy the mean can stay at zero. Your description of the Dirac delta as the 'limit' of the density function at 0 is very clear. Thanks mathman.
 
  • #5
SW VandeCarr said:
Previously I posted a question on the Dirac delta function and was informed it was not a true function, but rather a distribution. However, I have to admit I still did not understand why its integral (neg inf to pos inf) is unity. I've thought about this and came up with the following:

Consider a normal distribution with a variance of 0. Since the integral of a normal distribution is constrained to be unity, the mean must go to infinity as the variance approaches zero. This is the Dirac delta. Is this reasonable or am I wandering in the wilderness?


Yes the idea is okay, but since you cannot consider the dirac-delta as a function, you cannot either interpret the integral in its usual way. Basically, the idea is that the delta can be seen as a limit of a sequence of functions (also called distribution). Typically, that function can be a gaussian (but they are other equally acceptable definitions, in terms of sine for example). A gaussian depends upon x and sigma. If you choose to tends sigma to zero, since the gaussian is normalized (that is, overal area=1), the width tends to zero, the central peak tends to infinity, while everwhere else it tends to zero. Now, if you integrate such a gaussian multiplied by any function, then apply the limit for sigma tending to zero, the value of the function can be considered as a constant upon the withd, so, if the with is centered around zero, you can put the f(0) out of the integral, and because the gaussian is normalize, the integral gives you 1, so it remains just f(0). Hence the definition, at least from a heuristic (physics) point of view.
 
  • #6
For intuitions about the Dirac Delta, you should think of it as a bell curve with infinitesimal variance (spread). It's still got unit area under the curve, but all that area is concentrated in within an infinitesimal neighborhood of the mean.

EDIT. It seems you are thinking about this more or less correctly. And you're right, that the value at the mean is infinite (the inverse of an infinitesimal).
 

1. What is the Dirac delta distribution?

The Dirac delta distribution, also known as the Dirac delta function, is a mathematical concept used in the field of mathematics and physics. It is a theoretical function that is used to represent a point mass or impulse at a specific point. It is often used to simplify calculations in mathematical models and solve differential equations.

2. How is the Dirac delta distribution defined?

The Dirac delta distribution is defined as a function that has the value of 0 everywhere except at a single point, where it has an infinite value. It is often represented as δ(x) or δ(x-a), where a is the point at which the function has an infinite value. It is also defined as the limit of a sequence of functions that approach the Dirac delta function as a parameter approaches 0.

3. What are the properties of the Dirac delta distribution?

The Dirac delta distribution has several important properties, including the property of being "infinitely tall" at its single point of non-zero value, having an area under the curve of 1, and having a value of 0 everywhere else. It also satisfies the sifting property, which means that when integrated with a well-behaved function, it picks out the value of the function at the point of non-zero value.

4. How is the Dirac delta distribution used in physics?

The Dirac delta distribution is used in physics to model point particles, such as electrons, in quantum mechanics. It is also used in signal processing to represent impulses or sudden changes in a signal. In electromagnetics, it is used to describe the electric potential of a point charge. It is a powerful tool in many areas of physics and engineering.

5. What are some common misconceptions about the Dirac delta distribution?

There are several common misconceptions about the Dirac delta distribution, including that it is an actual function that exists in the physical world and that it can be graphed like a regular function. In reality, it is a theoretical concept that is used in mathematical models. It also does not have a value of infinity at its point of non-zero value, but rather an infinitely small value that is approaching infinity. Additionally, the Dirac delta distribution should not be confused with the Kronecker delta, which is a discrete version of the Dirac delta function.

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