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

• ## 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.