The CDF of the Sum of Independent Random Variables

In summary: But it works for mixed variables too.In summary, the conversation discusses two ways to find the CDF of a summation of independent and identically distributed random variables, either directly using convolutions or via Fourier transforms. It is also mentioned that the Laplace transform can be used, but the Fourier transform may be preferable in cases of infinite moments. The Fourier transform is similar to the Laplace transform, but is linked via a limit equation and the relationship in the complex plane. To get the CDF from the Fourier transform, one must obtain the PDF through the inverse transform and then integrate. A more general formula is also mentioned for when the variable can be discrete, continuous, or a mixture of both.
  • #1
EngWiPy
1,368
61
Hello all,

Suppose I have the following summation ##X=\sum_{k=1}^KX_k## where the ##\{X_k\}## are independent and identically distributed random variables with CDF and PDF of ##F_{X_k}(x)## and ##f_{X_k}(x)##, respectively. How can I find the CDF of ##X##?

Thanks in advance
 
Physics news on Phys.org
  • #3
mathman said:
There are two ways. Direct - using convolutions and via Fourier transforms.

https://www.statlect.com/fundamentals-of-probability/sums-of-independent-random-variables (direct)

The Fourier transform method. Get Fourier transform of each density function (in your case all the same), multiply together (in your case Kth power) and get the inverse Fourier transform.

Can I use the Laplace transform instead of the Fourier transform. I now remember in the past using the moment generating function (MGF). So, can I find the MGF of ##X##, and then find the inverse Laplace transform to find the CDF?
 
Last edited:
  • #4
You can use the Laplace transform as you described. The expressions for the Laplace and Fourier transforms are similar. My personal preference is Fourier, since the inverse transform seems easier.
 
  • Like
Likes EngWiPy
  • #5
He S_David.

I should point out that if the random variables are discrete random variables (as opposed to continuous ones) then you should look into probability generating functions.
 
  • Like
Likes EngWiPy
  • #6
chiro said:
He S_David.

I should point out that if the random variables are discrete random variables (as opposed to continuous ones) then you should look into probability generating functions.

Thanks. They are actually continuous random variables.
 
  • #7
One additional advantage to Fourier transform as opposed to Laplace is when moments are infinite. Example - Cauchy distribution.
 
  • Like
Likes EngWiPy
  • #8
mathman said:
One additional advantage to Fourier transform as opposed to Laplace is when moments are infinite. Example - Cauchy distribution.

The MGF of a random variable X is defined as

[tex]\mathcal{M}_X(s)=E\left[e^{-sX}\right]=\int_Xe^{-sX}f_X(x)\,dx[/tex]

In this definition we have the PDF of X is involved in the definition of the MGF. How the Fourier transform is similar to this? I mean, in the definition of the Fourier transform there is not PDF involved, right? How to get the CDF from the Fourier transform then?
 
  • #10
S_David said:
The MGF of a random variable X is defined as

[tex]\mathcal{M}_X(s)=E\left[e^{-sX}\right]=\int_Xe^{-sX}f_X(x)\,dx[/tex]

In this definition we have the PDF of X is involved in the definition of the MGF. How the Fourier transform is similar to this? I mean, in the definition of the Fourier transform there is not PDF involved, right? How to get the CDF from the Fourier transform then?
The limits of integration for a moment generating function are (in general) [itex](-\infty ,\infty )[/itex], so that it is possible the integral may not exist.

The Fourier transform is that of the PDF (similar to Laplace, except using [itex]e^{isx}[/itex]). To get CDF from Fourier transform, get PDF (using inverse transform) and integrate.
 
  • Like
Likes EngWiPy
  • #11
There is a more general formula for when the variable can be either discrete, continuous, or a mixture of the two (or even singular if you wish).

We have
[tex]P[X+Y\le x] = P[X\le x-Y] = E[F_X(x-Y)] = \int F_X(x-y)dF(y) [/tex]

The integral above is a Stieltjes integral so we recover the standard convolution formulas when X and Y are both purely discrete or purely absolutely continuous.
 

What is the CDF of the Sum of Independent Random Variables?

The CDF (cumulative distribution function) of the sum of independent random variables is a mathematical function that describes the probability that the sum of two or more independent random variables will be less than or equal to a certain value. This function takes into account the individual probability distributions of each random variable and calculates the overall probability of their sum.

How is the CDF of the Sum of Independent Random Variables calculated?

The CDF of the sum of independent random variables is calculated by convolving the individual probability distributions of each random variable. This involves taking the integral of the product of the individual CDFs. In simpler terms, it is the sum of the probabilities that each random variable will be less than or equal to the given value.

What is the relationship between the CDF of the Sum of Independent Random Variables and the individual CDFs?

The CDF of the sum of independent random variables is a combination of the individual CDFs. It takes into account the probabilities of each individual random variable and calculates the overall probability of their sum. This relationship is important in understanding the behavior of complex systems where multiple random variables are involved.

Can the CDF of the Sum of Independent Random Variables be used for any type of random variables?

Yes, the CDF of the sum of independent random variables can be used for any type of random variables, as long as they are independent. This means that the outcome of one random variable does not affect the outcome of another. This is a common assumption in many statistical models and allows for the use of the CDF of the sum of independent random variables in a wide range of applications.

What are some real-world applications of the CDF of the Sum of Independent Random Variables?

The CDF of the sum of independent random variables has many real-world applications, particularly in fields such as finance, engineering, and physics. It is commonly used in risk analysis, where the sum of multiple independent random variables (such as market fluctuations) can be used to calculate the overall risk of a system. It is also used in reliability analysis, where the failure rates of individual components are combined to determine the overall failure rate of a system.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
565
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
852
  • Set Theory, Logic, Probability, Statistics
2
Replies
36
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
16
Views
2K
  • Set Theory, Logic, Probability, Statistics
2
Replies
35
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
8
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
990
Back
Top