September 4, 2007 - Cosmic Telescope: Difference between revisions
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;Binomial distribtuion | ;Binomial distribtuion | ||
: best example is a coin toss, its either heads or tails | : best example is a coin toss, its either heads or tails | ||
: mean (<math>\mu</math>) = number of tries <math>n</math>(coin flips) * probability of success<math>p</math> (head, 1/2) | |||
:standard deviation(<math>\sigma</math>) = <math>np(1-p)</math> | |||
; Poisson Distribution | ; Poisson Distribution | ||
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: the distribtuion approximates the Binomial Distribution for the special case when the mean (<math>\mu</math>) is a lot less than the number of attempts to measure (<math>n</math>) because the probability of the event occurrring is small. | : the distribtuion approximates the Binomial Distribution for the special case when the mean (<math>\mu</math>) is a lot less than the number of attempts to measure (<math>n</math>) because the probability of the event occurrring is small. | ||
: In the cosmic ray telescope experiment the mean number of detected cosmic rays is much smaller than the number of cosmic rays passing by. | : In the cosmic ray telescope experiment the mean number of detected cosmic rays is much smaller than the number of cosmic rays passing by. | ||
;Gaussian/Normal Distribution | |||
:standard deviation<math>\sigma</math> = Half Width <math>\Gamma</math> / 2.354 | |||
:error = <math>2\ln(2) \sigma</math> | |||
{| border="1" cellpadding="20" cellspacing="0" | {| border="1" cellpadding="20" cellspacing="0" | ||
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Your instrumental uncertainty is approximately equal to the Poisson sigma (<math>\sqrt{138}= 11.7</math>) counts per hour. | Your instrumental uncertainty is approximately equal to the Poisson sigma (<math>\sqrt{138}= 11.7</math>) counts per hour. | ||
The cosmic ray telescope sounting experiment appears to be following Poisson statistics. | |||
Revision as of 16:07, 5 September 2007
| Plateau Zeroing | |||||||
|---|---|---|---|---|---|---|---|
| Time Start | Time Stop | Time elapsed (min.) | Thresholds on both Scintillators | Singles Count (on top) | Coinc. Counts | Coinc. per minute | Angle measure |
| 1:35 pm on 8/31 | 1:10 pm on 9/04 | 5725 min | 225 | 626816 | 1654 | .2889 | 75 degrees |
- Binomial distribtuion
- best example is a coin toss, its either heads or tails
- mean ([math]\displaystyle{ \mu }[/math]) = number of tries [math]\displaystyle{ n }[/math](coin flips) * probability of success[math]\displaystyle{ p }[/math] (head, 1/2)
- standard deviation([math]\displaystyle{ \sigma }[/math]) = [math]\displaystyle{ np(1-p) }[/math]
- Poisson Distribution
- standard deviation ([math]\displaystyle{ \sigma }[/math]) = root of the mean ([math]\displaystyle{ \sqrt{\mu} }[/math])
- use in counting experiments
- the distribtuion approximates the Binomial Distribution for the special case when the mean ([math]\displaystyle{ \mu }[/math]) is a lot less than the number of attempts to measure ([math]\displaystyle{ n }[/math]) because the probability of the event occurrring is small.
- In the cosmic ray telescope experiment the mean number of detected cosmic rays is much smaller than the number of cosmic rays passing by.
- Gaussian/Normal Distribution
- standard deviation[math]\displaystyle{ \sigma }[/math] = Half Width [math]\displaystyle{ \Gamma }[/math] / 2.354
- error = [math]\displaystyle{ 2\ln(2) \sigma }[/math]
| Time Start | Time Stop | Time elapsed (hour) | Singles Count (on top) | Coinc. Counts | Coinc. per hour | [math]\displaystyle{ \sqrt{N} }[/math] | [math]\displaystyle{ (x_i-\bar{x})^2 }[/math] |
| 1:20 pm | 2:20 pm | 1 | 5694 | 127 | 127 | 11.3 | 121 |
| 2:20 pm | 3:20 pm | 1 | 4896 | 136 | 136 | 11.6 | 4 |
| 3:20 pm | 4:20 pm | 1 | 4655 | 151 | 151 | 12.2 | 169 |
Mean [math]\displaystyle{ \mu }[/math] = 138 counts per hour
Instrumental Uncertainty =[math]\displaystyle{ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N-1}} = 12.12 }[/math] counts per hour
Your instrumental uncertainty is approximately equal to the Poisson sigma ([math]\displaystyle{ \sqrt{138}= 11.7 }[/math]) counts per hour.
The cosmic ray telescope sounting experiment appears to be following Poisson statistics.