# Confidence Interval - Poisson

Considering X ~ Poi(λ). Based on observations x1, ..., x100, iid and with the same distribution as X, we can estimate the confidence interval [0.3;1.8] for λ. Find the confidence interval, with the same coverage probability for P {X<=2}. TIP: express P {X<=2} in terms of λ

• Erdos
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Extend the deadline for a couple of hours if you can. I may be able to answer this.

• Manatuts
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Max. I can do is +1 hour, is it enough?

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• Manatuts
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Don't you need to calculate the probability coverage? From what I understood you would need to calculate the probability coverage for the first given interval and then use it to find another interval with P{X<=2} with the same probability coverage. How did you skip not finding it?

• Erdos
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What do you mean by probability coverage? You know that with certain confidence level lambda in inside the given interval. So with the same confidence level, P[x <=2] will lie in the interval I computed. There doesn't seem to be a need to compute anything else.

• Manatuts
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Sorry, some terms might get lost in translation. Coverage probability I mean the percentage of chance of lambda being inside the confidence interval. So I was thinking you would need to calculate the percentage of chance of lambda being inside [0,3;1,8] and use this same percentage to determine the other confidence interval for P[X>=2]

• Erdos
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No, that percentage of chance is not given for lambda, and hence you can not find it for p[X<=2]. That percentage of chance is unknown but is the same for both confidence intervals.

• Manatuts
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But how can you be certain that is the same for both?

• Erdos
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It is because the second interval is directly computed from the first interval, so the probability of both intervals are the same.

• Manatuts
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Makes sense. Why does he mention this coverage probability though? It seems like it wouldn't change your answer with your without it

• Erdos
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The problem does not ask for the coverage probability. It just wants it to be the same.