# Integration to d density function

• benoardo
In summary, the conversation is about a problem with calculating an integration involving the density function F(x) and the expectation. The conversation also discusses the concept of integrating with respect to different variables and the formula for the distribution of the sum of two independent random variables.

#### benoardo

I have a problem with an integration, namely:

Int(from 0 to x) (1-F(x-t)) dF(x)

and do not know how to calculate...:-(

If it help,s dF(x) = (dF/dx)dx

It helps, but that mean to multiply the integrand by the density f=dF/dx and then integrate over dx, so like how to derive the expectation...

but another problem is, that I do not know the density and need to have a result within F(x)

Well, you can do it, though you do'nt think you can. so let#'s play a littel game.

What's the integral of 1 with respect to x? 1 wrt y? 1 wrt F(x)?

Now what's the integral of x wrt x? y wrt y, F(s) wrt F(s)?

Note, I don't think you want an x in the integrand and in the limit.

That's almost all clear but in my book i read that F(s) wrt F(s) is something like F(s)*F(s) which stands here for the convolution...
And i have a x in the itegrand as well as in the limit.
As I said before: int(0 to x) 1-F(x-t) wrt F(t), sorry so you are write, it is F(t) not F(x)!

Thank you very much for your support, but yesterday i found the answer in my stochastic I scriptum...if someone is interested in:

Let x, y be independent rvs with dfs F and G then holds for the distribuntion of the sum x+y:

H(a)=P(x+y<=a)=int F(a-v)dG(v).

## 1. What is integration to density function?

Integration to density function is a mathematical process used to calculate the area under a probability density curve. It is commonly used in statistics and probability to calculate the probability of a certain event occurring within a given range of values.

## 2. How is integration to density function calculated?

Integration to density function is calculated by taking the integral of a probability density function over a given range of values. This can be done through numerical methods or by using calculus techniques such as the fundamental theorem of calculus.

## 3. What is the purpose of integration to density function?

The purpose of integration to density function is to calculate probabilities of events occurring within a given range of values. This can be useful in various fields such as statistics, finance, and engineering.

## 4. What are some applications of integration to density function?

Integration to density function has many applications in various fields, such as calculating the probability of stock prices reaching a certain level, estimating the likelihood of a disease occurring within a population, and determining the expected value of a continuous variable.

## 5. What are some common techniques used in integration to density function?

Some common techniques used in integration to density function include the trapezoidal rule, Simpson's rule, and Monte Carlo integration. These methods can be used to approximate the area under a curve and calculate the probability of events occurring within a range of values.