고려대학교 보건환경융합과학부 방사선안전분석 Lab 2 Statistics of Counting

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1. Poisson distribution

Poisson distribution은 시행 횟수는 아주 많으면서 성공 확률은 아주 낮은 경우 사용되는 확률 분포이며, N이 충분히 크고 p가 충분히 작아서 Np가 적당할 때 binomial distribution의 값을 근사적으로 구할 수 있습니다. Binomial distribution에서 Np=λ를 유지하면서 N→∞일 때, 그 분포는 Poisson distribution에 수렴합니다. Poisson distribution은 일반적으로 N≥20이고 p≤0.05이면 어느 정도 충분하고, N≥100이고 p<0.01이면 매우 훌륭하다고 할 수 있습니다.
2. Gaussian distribution

Poisson distribution에서 n ̅=Np≫1을 만족하면 Gaussian distribution을 유도할 수 있는데, Gaussian distribution의 기댓값과 분산이 Poisson distribution의 n ̅과 같다는 것을 이용합니다. Poisson distribution의 확률 질량 함수와 Gaussian distribution의 확률 밀도 함수가 서로 유사하다는 사실을 이용하여 Poisson distribution을 Gaussian distribution으로 근사할 수 있습니다. 이를 Poisson approximation이라고 합니다.

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1. Poisson distribution

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given that these events occur with a known constant rate and independently of the time since the last event. It is commonly used in various fields, such as biology, physics, and finance, to describe phenomena where events occur randomly and independently at a constant average rate. The Poisson distribution is characterized by a single parameter, the average rate of occurrence, which determines both the mean and the variance of the distribution. One of the key properties of the Poisson distribution is that the mean and variance are equal, which makes it a useful model for situations where the variance is proportional to the mean. Additionally, the Poisson distribution can be used to approximate the binomial distribution when the probability of success is small and the number of trials is large, which makes it a versatile tool for statistical analysis. Overall, the Poisson distribution is a fundamental concept in probability theory and has numerous applications in various scientific and practical domains.
2. Gaussian distribution

The Gaussian distribution, also known as the normal distribution, is a fundamental probability distribution in statistics and mathematics. It is characterized by a bell-shaped curve that is symmetric about its mean, and it is defined by two parameters: the mean and the standard deviation. The Gaussian distribution has several important properties that make it widely applicable in various fields. Firstly, it is a continuous probability distribution, which means it can be used to model continuous random variables, such as measurements or observations. Secondly, the Gaussian distribution has the property of being the limiting distribution of many other probability distributions, including the binomial, Poisson, and exponential distributions, under certain conditions. This property, known as the Central Limit Theorem, makes the Gaussian distribution a powerful tool for statistical inference and modeling. Additionally, the Gaussian distribution has the property of being the maximum entropy distribution, which means it maximizes the uncertainty or randomness of a random variable, given its mean and variance. This property makes the Gaussian distribution a natural choice for modeling many real-world phenomena, where the underlying processes are complex and subject to various sources of variation. Overall, the Gaussian distribution is a fundamental concept in probability theory and statistics, with widespread applications in fields such as physics, engineering, economics, and social sciences.

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