生物統計研讀筆記 - Probability Distribution 機率分佈

生物統計書籍內，描述間斷數據與連續數據的機率分佈概念

• Discrete Random Variables

  • Binomial Distribution

    • 4 assumptions:
      • Fixed number of n of trials 固定次數的試驗
      • Can be classified 2 conditions: success, failure
      • success prob: p, failure pro: 1-p
      • trials are independent between
    • Let X be a binomial random variable with parameters n and p.
      • $$\mu=E(X)=n p$$
      • $$\sigma_X^2=\operatorname{Var}(X)=n p(1-p)$$
      • Probability Density function $$f(x)=\frac{n !}{x !(n-x) !} p^x(1-p)^{n-x}$$

    • Poisson Approximation to the Binomial Distribution

      • n must be larger (at least 20)
      • p must be small (< 0.05)
      • approximation will be good if n >= 100, and np <= 10

    • Normal Approximation to the Binomial Distribution

      • np > 5
      • n(1-p) > 5

  • The Poisson Distribution

    • the number of event occurrences in a continuous interval of time or space
      • Ex: the number of radioactive decays in a time interval
      • Ex: the number of sedge plants per sampling quadrat
    • Assumptions:
      • Events occur one at a time
      • Two or more events do not occur precisely at the same moment or location.
      • The occurrence of an event in a given period is independent of the occurrence of an event in any previous or later non-overlapping period.
      • The expected number of events during any one period is the same as during any other period of the same length. This expected number of events is denoted by mu
    • Poisson random variable depends on the single parameter µ , unlike binomial distribution
    • Probability Density function: $$f(x)=\frac{e^{-\mu}(\mu)^x}{x !}$$

• Continuous Random Variables

  • The Normal Distribution

    • $$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^2 / 2 \sigma^2}$$
    • Where $\sigma$: standard deviation of the random variable
    • Where $\mu$ is its mean.