Lecture 5 probability model normal distribution binomial distribution xiaojinyu@seu.edu.cn
Lecture 5 probability model normal distribution & binomial distribution xiaojinyu@seu.edu.cn
Contents a normal distribution for continuous data a Binomial distribution for binary categorical data
Contents Normal distribution for continuous data Binomial distribution for binary categorical data 2
The normal distribution The most important distribution in statistics
The Normal Distribution The most important distribution in statistics
Normal distribution a Introduction to normal distribution ■ History ■ Parameters and shape standard normal distribution and z score ■ Area under the curve 日 Application Estimate of frequency distribution Reference interval (range)in health related field
4 Normal distribution Introduction to normal distribution ◼ History ◼ Parameters and shape ◼ standard normal distribution and Z score ◼ Area under the curve Application ◼ Estimate of frequency distribution ◼ Reference interval (range) in health_related field
HSTROY-NORMAL DISTRIBU a Johann carl friedrich gauss Germany a One of the greatest mathematician a Applied in physics, astronomy a Gaussian distribution (177~1855) AU656184200 177 DDR 6561842D0 20 Mark and Stamp in memory of Gauss
5 histroy-Normal Distribution Johann Carl Friedrich Gauss Germany One of the greatest mathematician Applied in physics, astronomy Gaussian distribution (1777~1855) Mark and Stamp in memory of Gauss
The Most Important Distribution a Many real life distributions are approximately normal. such as height, EFV1, weight, IQ, and so on. a Many other distributions can be almost normalized by appropriate data transformation(e.g. taking the log).When log X has a normal distribution, X is said to have a lognormal distribution
6 The Most Important Distribution Many real life distributions are approximately normal. such as height, EFV1,weight, IQ, and so on. Many other distributions can be almost normalized by appropriate data transformation (e.g. taking the log). When log X has a normal distribution, X is said to have a lognormal distribution
Frequency distributions of heights of adult men
7 Frequency distributions of heights of adult men. (a) (b) (c) (d)
Sample Population 日 Histogram a normal distribution curve 口 the area of the bars口 The area under the curve 日 Cumulative relative日 The cumulative probability frequency In the population 口 in the sample,the d Generally speaking the proportion of the boys chance that a boy of aged of age 12 that are lower 12 is lower than a than a specified height. specified height if he grow normall
8 Sample & Population Histogram- the area of the bars Cumulative relative frequency in the sample, the proportion of the boys of age 12 that are lower than a specified height. normal distribution curve The area under the curve The cumulative probability. In the population. Generally speaking, the chance that a boy of aged 12 is lower than a specified height if he grow normally
Definition of normal distribution 日X~N(1a3) X is distributed as normal distribution with mean u and variance O2 d The probability density function (PDF) f() for a normal distribution is given by (x f∫(X)= (-∞<X<+∞) G√2兀 Where: e=2.7182818285, base of natural logarithm T=3. 1415926536 ratio of the circumference of a circle to the diameter
9 Definition of Normal distribution X ～ N(, 2 ), X is distributed as normal distribution with mean and variance 2. The probability density function (PDF) f (x) for a normal distribution is given by Where: e = 2.7182818285, base of natural logarithm = 3.1415926536,ratio of the circumference of a circle to the diameter. X f X e 2 2 ( ) 2 1 ( ) 2 − − = (- < X < +)
The shape of a normal distribution (X-m)2 f(r) ∫(X) e √2丌 3 2 0 x 10
10 The shape of a normal distribution x 0 .1 .2 .3 .4 f(x) X f X e 2 2 ( ) 2 1 ( ) 2 − − =