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[object Object]
[object Object],[object Object]
Regular statement in statistics ,[object Object],[object Object],[object Object]
2.1  Explanation of Probability and Related Concepts ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Eg.  If the incidence rates of influenza in classes A, B and C  are 60% , 50% and 40% then   Odds: To measure risk Odds ratio: To compare risks
2.1.3  Bayes ’  formula ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
Conclusion: The risk of lung cancer for smoker is 5 times as much as that for ordinary people.
2.3 Binomial Distribution P ( white ball)=0.8 P (  yellow ball )=0.2 
In general, if : the  probability  of an event appearing in a trial  n   : times of  independently repeated  trials X   : random variable,  total times  of appearing such an event, then the probability of  X = x This variable X is called a  binomial variable , or say X following a  binomial distribution , denoted as  Why is it called Binomial? See following expansion:
2.3.2  Plot of Binomial Distribution
2.3.3  Population mean and population variance
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
2.4  Poisson Distribution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
It can be proved,  When  n ->∞, the  will tend to In general,  if the probability function of a random variable  X   has the above shape ,  then we say that this variable follows a  Poisson distribution  with parameter  ,  denoted by   .
Example : Red cell count on glass slide. Since   Divide the  glass slide   into  n  small grids  ----  big  n  , 0 or 1 ; P  (a red cell) =     ----  small probability ; With or without a cell  ----  independent ;  Therefore,   Number of cells ~ Poisson distribution
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
2.4.2 Plot of probability function , positive skew;  , approximately symmetric
Property of Poisson Distribution   ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],However
Example :   Five samples taken from a river , the number of colibacillus ( 大肠杆菌 )  were counted ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
2.5  Normal Distribution ,[object Object],[object Object],[object Object]
μ 1 μ 2 μ 3 Two parameters:  population mean  population variance Normal distribution denoted by
Standard normal distribution   ,  ,  To any normal variable  , after a transformation of standardization Z is called with  standardized normal deviate or  Z-value ,   or  Z-score
2.5.2  Area under the normal probability density curve ,[object Object],[object Object],[object Object],z 0
[object Object],Corresponding to 1.96,  the area of one tail is 0.025,  the area of two tails is 0.025   2 =   0.05
[object Object],Corresponding to 2.58,  the area of one tail is 0.005,  the area of two tails is 0.010  -2.58 Φ(-2.58)=0.005
 
Critical value  : Two sided critical value : One sided critical value
Distribution of  X 1 + X 2   still follow a normal distribution When  X 1  and  X 2  are independent,
2.5.3  Determination    of a reference range ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object]
Example  Based on the hemoglobin data of  120  healthy females ,  ,  ; and the histogram shows it  approximately follows a normal distribution.  Please estimate the two-sided 95% reference range for females.
Caution ,[object Object],[object Object],[object Object],[object Object]
2.5.4  Normal approximation of binomial distribution and Poisson distribution ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
 
 
Example  The infectious rate of hookworm( 钩虫 ) is  13% , if randomly select  150  people , what is the probability that  at  least  20   of them being infected ? The probability that at  least   20  of them being infected is 50% 。   Area of the rectangles on
[object Object],[object Object],[object Object],[object Object]
Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],5. Web resources http://statpages.org/
[object Object]
 

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Chapter 2 Probabilty And Distribution

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  • 7. Eg. If the incidence rates of influenza in classes A, B and C are 60% , 50% and 40% then Odds: To measure risk Odds ratio: To compare risks
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  • 10. Conclusion: The risk of lung cancer for smoker is 5 times as much as that for ordinary people.
  • 11. 2.3 Binomial Distribution P ( white ball)=0.8 P ( yellow ball )=0.2 
  • 12. In general, if : the probability of an event appearing in a trial n : times of independently repeated trials X : random variable, total times of appearing such an event, then the probability of X = x This variable X is called a binomial variable , or say X following a binomial distribution , denoted as Why is it called Binomial? See following expansion:
  • 13. 2.3.2 Plot of Binomial Distribution
  • 14. 2.3.3 Population mean and population variance
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  • 17. It can be proved, When n ->∞, the will tend to In general, if the probability function of a random variable X has the above shape , then we say that this variable follows a Poisson distribution with parameter , denoted by .
  • 18. Example : Red cell count on glass slide. Since Divide the glass slide into n small grids ---- big n , 0 or 1 ; P (a red cell) =  ---- small probability ; With or without a cell ---- independent ; Therefore, Number of cells ~ Poisson distribution
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  • 20. 2.4.2 Plot of probability function , positive skew; , approximately symmetric
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  • 25. μ 1 μ 2 μ 3 Two parameters: population mean population variance Normal distribution denoted by
  • 26. Standard normal distribution , , To any normal variable , after a transformation of standardization Z is called with standardized normal deviate or Z-value , or Z-score
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  • 31. Critical value : Two sided critical value : One sided critical value
  • 32. Distribution of X 1 + X 2 still follow a normal distribution When X 1 and X 2 are independent,
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  • 34.
  • 35. Example Based on the hemoglobin data of 120 healthy females , , ; and the histogram shows it approximately follows a normal distribution. Please estimate the two-sided 95% reference range for females.
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  • 39.  
  • 40. Example The infectious rate of hookworm( 钩虫 ) is 13% , if randomly select 150 people , what is the probability that at least 20 of them being infected ? The probability that at least 20 of them being infected is 50% 。 Area of the rectangles on
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  • 46.