Timeline for Estimating probability that a large sum of i.i.d variables is positive
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| May 8, 2019 at 8:58 | comment | added | Dieter Kadelka | @MattF. First I computed $N = 1000000$ realisations $(z_k)_{k=1}^N$ of $\sum_{i=1}^{100} Z_i$ and saw that $z_k < 0$ for each $k$. To exclude the possibitlty of a fault in my ad hoc program I counted the number of $z_k > -30$. This gave the estimation $P(Z > -30) \sim 0.00009$.Even this extimation is much smaller than $\Phi(-10/3)$ and $f(100)$ is even much smaller. Of course, having the values $z_1,\ldots,z_N$ it is possible to calculate a kernel density estimator of $\sum_{i=1}^{100} Z_i$, f.i. with R. | |
| May 7, 2019 at 16:28 | vote | accept | Bogdan | ||
| May 7, 2019 at 1:26 | comment | added | user44143 | @DieterKadelka, why are you looking at $P(Z<-30)$ to calculate $f(100)$? | |
| May 7, 2019 at 1:07 | answer | added | Iosif Pinelis | timeline score: 1 | |
| May 4, 2019 at 13:02 | comment | added | Dieter Kadelka | Sorry, the probability is $P(Z > -30) \sim 0.00009$. | |
| May 4, 2019 at 9:47 | comment | added | Dieter Kadelka | @Bogdan: I have simulated the problem for myself. The estimation of Matt F. seems to be useless for your problem. For instance with 1000000 runs I got an estimation of the probability $P(Z < -30) = 90/1000000 = 0.00009$, far away from $0.000429$. Confirming your obeservation, even with 1000000 runs there was no success for the original problem. | |
| May 4, 2019 at 8:17 | comment | added | Dieter Kadelka | @Bogdan: The estimation given by Matt F. seems to be very crude. The same sort of problems arise in estimating probabilities in risk theory. The technique of "Importance Sampling" (en.wikipedia.org/wiki/Importance_sampling) may be useful in your situation. | |
| May 3, 2019 at 18:00 | comment | added | user44143 | $Z$ has mean $-1$, standard deviation $3$, so $f(100) \sim \Phi(-10/3) \sim .000429$. | |
| May 3, 2019 at 16:38 | comment | added | user44143 | I edited the title to be consistent with the post. | |
| May 3, 2019 at 16:37 | history | edited | user44143 | CC BY-SA 4.0 |
edited title
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| May 3, 2019 at 16:32 | history | asked | Bogdan | CC BY-SA 4.0 |