History of Normal Distribution
Abraham de Moivre was the first one to discover the normal distribution. He developed the normal curve mathematically in 1733 as an approximation to the binomial distribution.
He found that when the number of trials in a binomial distribution gets larger, the shape of binomial distribution approaches a smooth curve.

正态分布：

Normal.png

The mean, median, and mode of the normal distribution are the same.

正态分布的例子：

1、Measurement errors usually follow a normal distribution with more times of small errors and less times of large errors.
2、The heights and weights of human and animal populations follow normal distribution.
3、Normal distributions are “bell” shaped, and symmetric.

正态分布的特征

所有的正态分布都可以被转化为标准的正态分布-Z转化

Z_transformation.png

Example of a Normal Distribution
Suppose we have a normal distribution of weights, with μ = 70 kg and σ = 10 kg.
(1)What is the probability of having a weight >80kg?

pnorm.png

R code: 1-pnorm(80,mean=70,sd=10)
(2)What is the probability of having a weight >60kg?

>60kg.png

R code:1-pnorm(60,mean=70,sd=10) #P(>60kg)=0.8413447
(3)What is the probability of having a weight 55~65kg?

55-65kg.png

R code: pnorm(65,mean=70,sd=10)-pnorm(55,mean=70,sd=10)

Example of a Normal Distribution
A group of medical genetics students investigate some patients with gastric cancer by checking the expression of RUNX3 gene in their gastric cancer cells. The mean and standard deviation of the expression are 0.25 and 0.07, respectively. What is the probability that a gastric cancer patient has a expression of that gene above 0.3?

Answer.png

Statistics with R

汇总.png

Format of common probability distribution in R

Format.png