共享（1）Poisson Distribution

delicatepage · 发表于 2004-7-11 03:51:40

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oisson Distribution
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align=left>Application
A random variable X has a Poisson distribution if the following conditions all hold:
<OL type=1>
<LI class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-pagination: widow-orphan; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt">The number of outcomes of a trial occurring in one time interval (or specified region of space) is INDEPENDENT of the number occurring in any other non-overlapping interval (or region) -- i.e. events occur at random in some continuous time or space. </LI>
<LI class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-pagination: widow-orphan; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt">The probability of a single outcome during a short time interval (or small region) is proportional to the length of the time interval (or size of the region) and does not depend on the number of events occurring outside of that interval. </LI>
<LI class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-pagination: widow-orphan; mso-list: l0 level1 lfo1; tab-stops: list 36.0pt">The probability that more than one outcome will occur in such a short time interval (or small region) is negligible. </LI></OL>
Distribution
If X is a Poisson random variable with parameter lambda, then the PMF, CDF, mean, and variance are as follows
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Examples
<OL type=1>
<LI class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-pagination: widow-orphan; mso-list: l1 level1 lfo2; tab-stops: list 36.0pt">The number of customers that will come into a business during a given time interval </LI>
<LI class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-pagination: widow-orphan; mso-list: l1 level1 lfo2; tab-stops: list 36.0pt">The number of weeds in 1 square meter of a corn field. </LI>
<LI class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-pagination: widow-orphan; mso-list: l1 level1 lfo2; tab-stops: list 36.0pt">The number of white blood cells in 1 milliliter of blood. </LI></OL>
Relationships to Other Distributions
<UL type=disc>
<LI class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-pagination: widow-orphan; mso-list: l2 level1 lfo3; tab-stops: list 36.0pt">The Poisson distribution can be used to approximate the Binomial distribution if n is large and p is sufficiently near zero. </LI>
<LI class=MsoNormal style="MARGIN: 0cm 0cm 0pt; TEXT-ALIGN: left; mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; mso-pagination: widow-orphan; mso-list: l2 level1 lfo3; tab-stops: list 36.0pt">In a "Poisson process", the intervals between successive events have independent and identically distributed Exponential distributions (with parameter lambda) while the number of events in a specified interval has the Poisson distribution .</LI></UL>

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