The CDF from the characteristic function

Click For Summary
SUMMARY

The discussion focuses on deriving the cumulative distribution function (CDF) of the random variable \( Y = \sum_{k=1}^K \frac{a_k}{b_k} \), where \( \{a_k, b_k\} \) are independent and identically distributed (i.i.d.) exponential random variables with parameter 1. The moment generating function (MGF) is deemed unsuitable due to undefined means, leading to the use of the characteristic function (CF) instead. The CF is derived as \( \psi_Y(t) = \left[1 + je^{-jt}E_1(-jt)\right]^K \). The CDF is attempted using the Gil-Pelaez theorem, but numerical integration yields results outside the expected range of [0, 1], indicating potential errors in the integral evaluation.

PREREQUISITES
  • Understanding of characteristic functions (CF) in probability theory
  • Familiarity with moment generating functions (MGF) and their limitations
  • Knowledge of numerical integration techniques for evaluating integrals
  • Proficiency in using mathematical software like Mathematica for symbolic computation
NEXT STEPS
  • Learn about the Gil-Pelaez theorem for inverting characteristic functions to obtain CDFs
  • Study numerical integration methods to ensure accurate results in MATLAB and Mathematica
  • Explore the properties and applications of the exponential integral function \( E_1(x) \)
  • Investigate the implications of undefined means in random variable transformations
USEFUL FOR

Mathematicians, statisticians, and data scientists working on probability distributions, particularly those dealing with characteristic functions and numerical integration challenges.

  • #31
Look at post #27. \int_0^{\infty}\frac{sin(tx)}{t}dt=0,\ for\ x=0
 
Physics news on Phys.org
  • #32
Ooops! I forgot about that. My mistake. Thanks!

So, this is an indication that the derivation is correct, right? The only difference in my equations is that I use the formula ##\text{ci}(x)\pm j\,\text{si}(x) = \text{Ei}(\pm j\,x)##, where ##\text{Ei}(x) = -\int_{-x}^{\infty}\frac{e^{-t}}{t}\,dt = -E_1(-x)##. The question is: why the numerical integration doesn't evaluate as a CDF for the general case using MATLAB and Mathematica? Is it a precision issue in the software tools?
 
  • #33
I don't know how MATLAB or Mathematica were set up. There may be some limitations to their applicability. Since I have never used them I can't help here. When I need complicated integrals I look up the table I previously mentioned.
 
  • #34
mathman said:
I don't know how MATLAB or Mathematica were set up. There may be some limitations to their applicability. Since I have never used them I can't help here. When I need complicated integrals I look up the table I previously mentioned.

But theoretically, the derivation should be correct, right? Because I just raise the CF of a single X to the power K, and then solve for the CDF as we did for a single X.
 
  • #35
The basic theory (invert the nth power of the char. function gives the cdf of the sum of n terms) is correct. But as we have seen with the case n=1, the calculation can be very tricky. The Gil-Pelaez theorem had to be used and getting the imaginary part can be messy.
 
  • #36
Yes, but supposedly, the software tools take care of the imaginary part using their functions to extract the imaginary part. We don't have to concern ourselves with doing that manually.
 
  • #37
I can't comment on these tools, since I've never used them. If you are sure they should work, I suggest you check the input if the answer doesn't look right.
 

Similar threads

Undergrad Relation with Hessian and Log-likelihood
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad The CDF from the Characteristic Function
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad CDF of summation of random variables
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad The characteristic function of order statistics
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Finding the derivative of a characteristic function
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad QED replacing photon field with current in 3-point function
  • · Replies 1 ·
Replies
1
Views
1K
MHB Markov Process Limit: Calculating $u_k$ as a Function of $a,b$
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Characteristic functions and functionals
  • · Replies 9 ·
Replies
9
Views
2K
Graduate Conditional expectation and covariance of function of RVs
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad Proving a multivariate normal distribution by the moment generating function
  • · Replies 5 ·
Replies
5
Views
3K