东南大学：《数学建模》课程教学资源（PPT课件讲稿）Lecture 5 概率模型 probability model（normal distribution & binomial distribution）

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     − − =