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How to fit Poission distribution with large mean?
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I have one set of discrete data plotted below. I suspect the data follows Poisson distribution with mean $mu simeq 400$ and standard deviation $sigma simeq sqrt{400} = 20$.
$$ n(k) = frac{mu ^k}{k!} exp (- mu) (mathrm{eq}.1)$$
But how can I confirm this? I tried numerically to calculate (eq.1) but, as you expect, it overflowed.
statistics probability-distributions data-analysis
asked Jan 13 at 6:56
ynnynn
616
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add a comment |
$begingroup$
I have one set of discrete data plotted below. I suspect the data follows Poisson distribution with mean $mu simeq 400$ and standard deviation $sigma simeq sqrt{400} = 20$.
$$ n(k) = frac{mu ^k}{k!} exp (- mu) (mathrm{eq}.1)$$
But how can I confirm this? I tried numerically to calculate (eq.1) but, as you expect, it overflowed.
statistics probability-distributions data-analysis
asked Jan 13 at 6:56
ynnynn
616
$endgroup$
add a comment |
$begingroup$
I have one set of discrete data plotted below. I suspect the data follows Poisson distribution with mean $mu simeq 400$ and standard deviation $sigma simeq sqrt{400} = 20$.
$$ n(k) = frac{mu ^k}{k!} exp (- mu) (mathrm{eq}.1)$$
But how can I confirm this? I tried numerically to calculate (eq.1) but, as you expect, it overflowed.
statistics probability-distributions data-analysis
asked Jan 13 at 6:56
ynnynn
616
$endgroup$
I have one set of discrete data plotted below. I suspect the data follows Poisson distribution with mean $mu simeq 400$ and standard deviation $sigma simeq sqrt{400} = 20$.
$$ n(k) = frac{mu ^k}{k!} exp (- mu) (mathrm{eq}.1)$$
But how can I confirm this? I tried numerically to calculate (eq.1) but, as you expect, it overflowed.
statistics probability-distributions data-analysis
statistics probability-distributions data-analysis
asked Jan 13 at 6:56
ynnynn
616
asked Jan 13 at 6:56
ynnynn
616
asked Jan 13 at 6:56
ynnynn
616
asked Jan 13 at 6:56
ynnynn
616
asked Jan 13 at 6:56
ynnynn
616
616
add a comment |
add a comment |
1 Answer
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Central Limit theorem - a Poisson distribution with a large mean is extremely similar to a discretized normal. With standard deviation $20$, each of those integer points will represent a $z$-score range of $.05$ - from $-0.025$ to $0.025$ at $400$, from $0.025$ to $0.075$ at $401$, and so on. The peak at $400$ should be a probability of about $frac1{20sqrt{2pi}}approx 0.02$.
You're getting numbers that are nearly twice that. Empirically, it looks like your hypothesized model is wrong, and the data is clustered tighter than a Poisson distribution would be.
answered Jan 13 at 7:06
jmerryjmerry
8,3781022
$endgroup$
$begingroup$
Thank you. It helped me a lot. I have to reconstruct the model though.
$endgroup$
– ynn
Jan 13 at 8:39
$begingroup$
On that, there's a quantitative way to check - you should have enough data there to estimate the variance.
$endgroup$
– jmerry
Jan 13 at 8:55
$begingroup$
I repeated the calculation 500 times but the result didn't change. After some research, it turned out that the model (actually this is Erdos-Renyi random graph model) was not wrong but my estimation "it should always follow Poisson distribution" were incorrect. Now I have what I wanted. Thank you again.
$endgroup$
– ynn
Jan 13 at 10:27
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
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$begingroup$
Central Limit theorem - a Poisson distribution with a large mean is extremely similar to a discretized normal. With standard deviation $20$, each of those integer points will represent a $z$-score range of $.05$ - from $-0.025$ to $0.025$ at $400$, from $0.025$ to $0.075$ at $401$, and so on. The peak at $400$ should be a probability of about $frac1{20sqrt{2pi}}approx 0.02$.
You're getting numbers that are nearly twice that. Empirically, it looks like your hypothesized model is wrong, and the data is clustered tighter than a Poisson distribution would be.
answered Jan 13 at 7:06
jmerryjmerry
8,3781022
$endgroup$
$begingroup$
Thank you. It helped me a lot. I have to reconstruct the model though.
$endgroup$
– ynn
Jan 13 at 8:39
$begingroup$
On that, there's a quantitative way to check - you should have enough data there to estimate the variance.
$endgroup$
– jmerry
Jan 13 at 8:55
$begingroup$
I repeated the calculation 500 times but the result didn't change. After some research, it turned out that the model (actually this is Erdos-Renyi random graph model) was not wrong but my estimation "it should always follow Poisson distribution" were incorrect. Now I have what I wanted. Thank you again.
$endgroup$
– ynn
Jan 13 at 10:27
add a comment |
$begingroup$
Central Limit theorem - a Poisson distribution with a large mean is extremely similar to a discretized normal. With standard deviation $20$, each of those integer points will represent a $z$-score range of $.05$ - from $-0.025$ to $0.025$ at $400$, from $0.025$ to $0.075$ at $401$, and so on. The peak at $400$ should be a probability of about $frac1{20sqrt{2pi}}approx 0.02$.
You're getting numbers that are nearly twice that. Empirically, it looks like your hypothesized model is wrong, and the data is clustered tighter than a Poisson distribution would be.
answered Jan 13 at 7:06
jmerryjmerry
8,3781022
$endgroup$
$begingroup$
Thank you. It helped me a lot. I have to reconstruct the model though.
$endgroup$
– ynn
Jan 13 at 8:39
$begingroup$
On that, there's a quantitative way to check - you should have enough data there to estimate the variance.
$endgroup$
– jmerry
Jan 13 at 8:55
$begingroup$
I repeated the calculation 500 times but the result didn't change. After some research, it turned out that the model (actually this is Erdos-Renyi random graph model) was not wrong but my estimation "it should always follow Poisson distribution" were incorrect. Now I have what I wanted. Thank you again.
$endgroup$
– ynn
Jan 13 at 10:27
add a comment |
$begingroup$
Central Limit theorem - a Poisson distribution with a large mean is extremely similar to a discretized normal. With standard deviation $20$, each of those integer points will represent a $z$-score range of $.05$ - from $-0.025$ to $0.025$ at $400$, from $0.025$ to $0.075$ at $401$, and so on. The peak at $400$ should be a probability of about $frac1{20sqrt{2pi}}approx 0.02$.
You're getting numbers that are nearly twice that. Empirically, it looks like your hypothesized model is wrong, and the data is clustered tighter than a Poisson distribution would be.
answered Jan 13 at 7:06
jmerryjmerry
8,3781022
$endgroup$
Central Limit theorem - a Poisson distribution with a large mean is extremely similar to a discretized normal. With standard deviation $20$, each of those integer points will represent a $z$-score range of $.05$ - from $-0.025$ to $0.025$ at $400$, from $0.025$ to $0.075$ at $401$, and so on. The peak at $400$ should be a probability of about $frac1{20sqrt{2pi}}approx 0.02$.
You're getting numbers that are nearly twice that. Empirically, it looks like your hypothesized model is wrong, and the data is clustered tighter than a Poisson distribution would be.
answered Jan 13 at 7:06
jmerryjmerry
8,3781022
answered Jan 13 at 7:06
jmerryjmerry
8,3781022
answered Jan 13 at 7:06
jmerryjmerry
8,3781022
answered Jan 13 at 7:06
jmerryjmerry
8,3781022
8,3781022
$begingroup$
Thank you. It helped me a lot. I have to reconstruct the model though.
$endgroup$
– ynn
Jan 13 at 8:39
$begingroup$
On that, there's a quantitative way to check - you should have enough data there to estimate the variance.
$endgroup$
– jmerry
Jan 13 at 8:55
$begingroup$
I repeated the calculation 500 times but the result didn't change. After some research, it turned out that the model (actually this is Erdos-Renyi random graph model) was not wrong but my estimation "it should always follow Poisson distribution" were incorrect. Now I have what I wanted. Thank you again.
$endgroup$
– ynn
Jan 13 at 10:27
add a comment |
$begingroup$
Thank you. It helped me a lot. I have to reconstruct the model though.
$endgroup$
– ynn
Jan 13 at 8:39
$begingroup$
On that, there's a quantitative way to check - you should have enough data there to estimate the variance.
$endgroup$
– jmerry
Jan 13 at 8:55
$begingroup$
I repeated the calculation 500 times but the result didn't change. After some research, it turned out that the model (actually this is Erdos-Renyi random graph model) was not wrong but my estimation "it should always follow Poisson distribution" were incorrect. Now I have what I wanted. Thank you again.
$endgroup$
– ynn
Jan 13 at 10:27
$begingroup$
Thank you. It helped me a lot. I have to reconstruct the model though.
$endgroup$
– ynn
Jan 13 at 8:39
$begingroup$
Thank you. It helped me a lot. I have to reconstruct the model though.
$endgroup$
– ynn
Jan 13 at 8:39
$begingroup$
On that, there's a quantitative way to check - you should have enough data there to estimate the variance.
$endgroup$
– jmerry
Jan 13 at 8:55
$begingroup$
On that, there's a quantitative way to check - you should have enough data there to estimate the variance.
$endgroup$
– jmerry
Jan 13 at 8:55
$begingroup$
I repeated the calculation 500 times but the result didn't change. After some research, it turned out that the model (actually this is Erdos-Renyi random graph model) was not wrong but my estimation "it should always follow Poisson distribution" were incorrect. Now I have what I wanted. Thank you again.
$endgroup$
– ynn
Jan 13 at 10:27
$begingroup$
I repeated the calculation 500 times but the result didn't change. After some research, it turned out that the model (actually this is Erdos-Renyi random graph model) was not wrong but my estimation "it should always follow Poisson distribution" were incorrect. Now I have what I wanted. Thank you again.
$endgroup$
– ynn
Jan 13 at 10:27
add a comment |
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