What is the Mean of Exponential RVs with Chi Square Distribution?

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Discussion Overview

The discussion revolves around deriving the mean of exponential random variables that follow a chi-square distribution with degrees of freedom 2n. Participants explore various approaches to understand the relationship between exponential and chi-square distributions, particularly in the context of sample means and moment-generating functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants express uncertainty about deriving the mean of exponential random variables related to the chi-square distribution with degrees of freedom 2n.
  • One participant suggests using the moment-generating function (MGF) to calculate the mean, noting that it exists for the exponential distribution.
  • Another participant points out that the sum of exponential random variables results in a gamma random variable, and questions how to derive a chi-square distribution from averaging gamma variables.
  • It is mentioned that the exponential distribution is a special case of the gamma distribution, specifically when k=1, and that the sample means may asymptotically approach a chi-square distribution with a mean of 2 when lambda=1/2.
  • There is a clarification regarding the nature of hyper-exponential distributions and their relation to the gamma distribution.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method to derive the mean or the relationship between the distributions. Multiple competing views and approaches remain, with some participants expressing confusion and others providing differing insights.

Contextual Notes

There are unresolved assumptions regarding the derivation process and the specific conditions under which the relationships between the distributions hold. The discussion includes various mathematical steps that are not fully resolved.

zli034
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I forgot how to derive the mean of exponential random variables follow the chi square distribution with degree freedom 2n. Don't know where I got it wrong. Anyone have a clue how to do it?

Thanks
 
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zli034 said:
I forgot how to derive the mean of exponential random variables follow the chi square distribution with degree freedom 2n. Don't know where I got it wrong. Anyone have a clue how to do it?

Thanks

[tex]\mu = \int_0^{\infty}t f(t)dt[/tex] where [tex]f(t)[/tex] is the pdf.
 
I want the distribution of the sample mean, which should has a chi square distribution. I know how to do the expectation of the sample mean, which is u.

Not quite what I am asking for. Thanks for the reply
 
try calculating the moment-generating function of the mean (it exists, since all moments of the exponential distribution exist). Note that for ANY distribution, where all the following integrals exist, the moment generating function of the sample mean is

[tex] m_{\overline x} (s) = E[e^{s \frac 1 n \sum x}] = E[\prod e^{(\frac s n)x}] = \prod{E[e^{(\frac s n)x}]}[/tex]

and each factor in the final product is calculated based on the exponential distribution.
 
I don't get it by doing MGF. Sum of exponential RVs is a Gamma RV. Chi square is the special case of Gamma with parameter is 1/2. I don't know how to from a Gamma by taking average to get the Chi.
 
zli034 said:
I don't get it by doing MGF. Sum of exponential RVs is a Gamma RV. Chi square is the special case of Gamma with parameter is 1/2. I don't know how to from a Gamma by taking average to get the Chi.

Well the exponential distribution is a special case of the gamma dist. where k=1. In addition, when lambda=1/2, I believe the distribution of the sample means asymptotically approaches a chi square dist. with a mean of 2 (2 degrees of freedom). If so, what exactly is your question?
 
SW VandeCarr said:
Well the exponential distribution is a special case of the gamma dist. where k=1. In addition, when lambda=1/2, I believe the distribution of the sample means asymptotically approaches a chi square dist. with a mean of 2 (2 degrees of freedom). If so, what exactly is your question?

If you're asking about a hyper-exponential dist.; it is still a one parameter dist., not a higher form (k>1) of the gamma dist.
 

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