고려대학교 보건환경융합과학부 방사선안전분석 Final Report

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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 있다. Binomial distribution에서 Np=λ를 유지하면서 N→∞일 때, 그 분포는 Poisson distribution에 수렴한다. λt≪1,N≫1 and Np≪N^(1/2)일 때 Binomial distribution은 근사식을 이용해 Poisson distribution으로 쉽게 바꿀 수 있다.

• 2. Gaussian distribution

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

• 3. Binomial distribution

  Binomial distribution은 N번의 독립 베르누이 시행(한 번의 시행에서 결과가 O or X 로 나오는 시행)에서 성공 확률이 p일 때 확률 분포이다. N번의 시행 중 n개가 성공일 확률 W(n)은 다음과 같다.

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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 to model rare events, such as the number of customers arriving at a store per hour or the number of radioactive decays in a given time period. The Poisson distribution is characterized by a single parameter, λ, which represents the average number of events occurring in the given interval. The probability mass function of the Poisson distribution is given by P(X=x) = e^(-λ) * (λ^x) / x!, where x is the number of events and e is the base of the natural logarithm. The Poisson distribution is useful in a variety of applications, including queuing theory, epidemiology, and reliability engineering. It is an important tool in statistical analysis and has many practical applications in various fields.

• 2. Gaussian distribution

  The Gaussian distribution, also known as the normal distribution, is a fundamental probability distribution in statistics and probability theory. It is a continuous probability distribution that is symmetric about its mean, with a bell-shaped curve. The Gaussian distribution is characterized by two parameters: the mean (μ) and the standard deviation (σ). The probability density function of the Gaussian distribution is given by f(x) = (1 / (σ * √(2π))) * e^(-(x-μ)^2 / (2σ^2)), where x is the random variable. The Gaussian distribution is widely used in various fields, including physics, engineering, economics, and social sciences, due to its many desirable properties. It is the basis for many statistical techniques, such as hypothesis testing, regression analysis, and time series analysis. The central limit theorem also states that the sum of a large number of independent random variables will converge to a Gaussian distribution, making it a crucial concept in probability and statistics.

• 3. Binomial distribution

  The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes (success or failure) with a constant probability of success. The binomial distribution is characterized by two parameters: the number of trials (n) and the probability of success in each trial (p). The probability mass function of the binomial distribution is given by P(X=x) = (n choose x) * p^x * (1-p)^(n-x), where x is the number of successes and (n choose x) is the binomial coefficient. The binomial distribution is widely used in various fields, such as quality control, market research, and clinical trials, to model the number of successes in a fixed number of independent trials. It is an important concept in probability and statistics, as it forms the basis for many other probability distributions and statistical techniques, such as the Poisson distribution and the normal approximation to the binomial distribution.