Timeline for How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2014 at 17:11 | comment | added | esg | In my view there is no need to cite this. If you think it is appropriate just cite this discussion (weblink). | |
| Nov 18, 2014 at 2:10 | comment | added | hengxin | I have got it. Thanks a lot. I want to use your idea about this calculation in my working paper. Then how should I cite it if you don't mind? | |
| Nov 17, 2014 at 19:55 | comment | added | esg | By Fubini's theorem you proceed as follows: (1) in the first step consider the values $0\leq r_m < r_{m+1}$ of $R_m$ resp. $R_{m+1}$ as fixed and integrate with resp. to $L$, you get $\mathbb{P}(r_m\leq L < r_{m+1})=\exp(-\lambda r_m)- \exp(-\lambda r_{m+1})=:f(r_m,r_{m+1})$ (2) in the second step you integrate $f$ with resp. to $(R_m,R_{m+1}$ | |
| Nov 14, 2014 at 14:50 | vote | accept | hengxin | ||
| Nov 11, 2014 at 13:00 | comment | added | hengxin | Could you please give me more hints on the first step of "taking expectation w.r.t $L$", namely $$\mathbb{P}(R_m\leq L < R_{m+1})=\mathbb{E}(e^{-\lambda R_m}-e^{-\lambda R_{m+1}})$$ Thanks. | |
| Jun 6, 2014 at 14:18 | comment | added | hengxin | It is not easy for me to follow your idea now. I have to learn the material first before giving some useful feedback. Thanks for your help. | |
| Jun 6, 2014 at 13:52 | comment | added | esg | 1) Yes, I am referring to Fubini's theorem (as it is formulated in the link you give. It allows (under appropriate conditions) to compute a multiple integral as an iterate integral) (2) if $L$ is independent of $(R_m,R_{m+1})$ and $0\leq R_m \leq R_{m+1}$ the first step above is valid (3) thus you only have to compute $\mathbb{E}e^{-\lambda R_m}$ resp.$\mathbb{E}e^{-\lambda R_{m+1}}$ (4) for your case (independent summands) it's easy: just use the product rule for the expectation of a product independent rvs | |
| Jun 5, 2014 at 12:54 | comment | added | hengxin | I have borrowed the book by W.Feller from our library. It is hard for me to find where the Fubini theorem is (in the Chinese translation version). The Laplace-transform can be found easily in the table of contents. Are you referring to [Fubini's theorem [wiki]](en.wikipedia.org/wiki/Fubini's_theorem)? In addition, is your approach general enough to handle with another similar probability: $$\mathbb{P} \big(\sum_{i=1}^{m} S_i + \sum_{i=1}^{m-1} A_i \le L < \sum_{i=1}^{m+1} S_i + \sum_{i=1}^{m} A_i \big)$$ where the numbers of $S_i$ and $A_i$ differ by one? Thanks. | |
| May 27, 2014 at 18:16 | comment | added | esg | (1) For the first step I've only used Fubini's theorem. (2) The Laplace-transform of a (nonnegative) rv $X$ is just the function $p\mapsto \mathbb{E}e^{-pX}$. (3) You can find both e.g. in W.Feller, An Intro. to Prob. Theory 2, (1970) | |
| May 27, 2014 at 3:34 | comment | added | hengxin | Thanks for your neat answer avoiding the messy calculation of $\Gamma(m, \lambda) + \Gamma(m, \mu)$. Basically it is beyond my current knowledge. Could you please offer me more details or references to the techniques you used such as "Laplace-transform" and "taking expectation w.r.t. $L$"? | |
| May 26, 2014 at 19:32 | history | answered | esg | CC BY-SA 3.0 |