Explain poisson distribution with their properties and uses
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heya ●_●
●●●MARK AS BRAINLIEST●●●
In probability theory and statistics, the Poisson distribution named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event.The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume
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The expected value and variance of a Poisson-distributed random variable are both equal to λ.The coefficient of variation is {\displaystyle \textstyle \lambda ^{-1/2}}, while the index of dispersion is 1.[5]The mean absolute deviation about the mean is[5]{\displaystyle \operatorname {E} |X-\lambda |=2\exp(-\lambda ){\frac {\lambda ^{\lfloor \lambda \rfloor +1}}{\lfloor \lambda \rfloor !}}.}The mode of a Poisson-distributed random variable with non-integer λ is equal to {\displaystyle \scriptstyle \lfloor \lambda \rfloor }, which is the largest integer less than or equal to λ. This is also written as floor(λ). When λ is a positive integer, the modes are λ and λ − 1.All of the cumulants of the Poisson distribution are equal to the expected value λ. The nth factorial moment of the Poisson distribution is λn.The expected value of a Poisson process is sometimes decomposed into the product of intensity and exposure (or more generally expressed as the integral of an "intensity function" over time or space, sometimes described as “exposure”
●●●MARK AS BRAINLIEST●●●
In probability theory and statistics, the Poisson distribution named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event.The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume
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The expected value and variance of a Poisson-distributed random variable are both equal to λ.The coefficient of variation is {\displaystyle \textstyle \lambda ^{-1/2}}, while the index of dispersion is 1.[5]The mean absolute deviation about the mean is[5]{\displaystyle \operatorname {E} |X-\lambda |=2\exp(-\lambda ){\frac {\lambda ^{\lfloor \lambda \rfloor +1}}{\lfloor \lambda \rfloor !}}.}The mode of a Poisson-distributed random variable with non-integer λ is equal to {\displaystyle \scriptstyle \lfloor \lambda \rfloor }, which is the largest integer less than or equal to λ. This is also written as floor(λ). When λ is a positive integer, the modes are λ and λ − 1.All of the cumulants of the Poisson distribution are equal to the expected value λ. The nth factorial moment of the Poisson distribution is λn.The expected value of a Poisson process is sometimes decomposed into the product of intensity and exposure (or more generally expressed as the integral of an "intensity function" over time or space, sometimes described as “exposure”
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