Timeline for There is mathematics behind the 1989 Tour de France !
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Jan 6, 2011 at 18:09 | comment | added | B R | I wouldn't be too concerned: naive simulations of rare events should not be taken too literally. | |
| Jan 6, 2011 at 16:51 | comment | added | Douglas Zare | Actually, that $0.004$ was the probability Greg's real time was at least $8/2=4$ seconds more than the sum of his rounded times. The actual value is $P(X_{22}\gt 19).$ If I calculate correctly, that is $122221817/4390627842881280000$ $=2.7837 ~10^{−11},$ although I am concerned that this does not agree with BR's simulation. The sum of the errors would have to be about $5.9$ standard deviations away from the mean, and the normal approximation would overestimate the size of the tail in this region. | |
| Jan 6, 2011 at 14:25 | history | edited | Alex B. | CC BY-SA 2.5 |
Corrected by considering difference distributions; deleted 2 characters in body; added 185 characters in body
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| Jan 6, 2011 at 14:20 | comment | added | Tony Huynh | As I understand it, $f_X$ gives us the distribution of the error just for a single rider. So Kevin's integral (assuming we take the floor, say) gives us the probability that Greg's real time was at least 8 seconds more than Greg's recorded time. But what we really want is the probability that Greg's real time is more than Laurent's real time. This will be a lot less than 0.004. | |
| Jan 6, 2011 at 14:08 | comment | added | Denis Serre | Isn't $f_X$ the self-convolution of $n$ copies of the uniform probability over $(0,1)$ ? | |
| Jan 6, 2011 at 13:58 | comment | added | Tony Huynh | @Kevin: I'm not sure that this is the integral that we want though. The respective roundoffs for Greg and Laurent are assumed to be uniform, but the difference between the roundoffs is certainly not uniform. See my answer below. | |
| Jan 6, 2011 at 13:52 | comment | added | Kevin O'Bryant | Mathematica says (with $n=11$) that $\int_8^{11} f_X(x)\,dx$ is $\frac{8593}{2217600} \approx 0.004$. | |
| Jan 6, 2011 at 13:46 | history | edited | Kevin O'Bryant | CC BY-SA 2.5 |
fixed link
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| Jan 6, 2011 at 12:45 | history | answered | Alex B. | CC BY-SA 2.5 |