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Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
Chapter 2 Probabilty And Distribution
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Chapter 2 Probabilty And Distribution

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  • 1.
    • Chapter-2
  • 2.
    • Chapter 2
    • Probability and Distribution
  • 3. Regular statement in statistics
    • Two parts :
    • Conclusion
    • Probability that the conclusion is true
  • 4. 2.1 Explanation of Probability and Related Concepts
    • 2.1.1 Probability
    • Flipping a die ( 骰子 )
    • Possible outcome: 1, 2, …, 6
    • Probability of “1”= 1/6
    • Color blindness test
    • Possible outcome: normal, abnormal
    • Probabilities of “abnormal”= ? ---- Unknown!
    • Survey: Randomly selected n students,
    • if m of them are color blinders, then
    • probability of abnormal 
  • 5.
    • In general,
    • Events: the possible outcomes ,…
    • Probability of the event E : P ( E ).
    • ---- between 0 and 1
    • Conditional probability :
    • Under the condition that appears, the
    • probability of the event
    • For example,
    • P ( nasopharyngeal carcinoma∣ EB virus +)
  • 6.
    • 2.1.2 Odds
    • Complementary event : If there are only two
    • possible events and they are exclusive, denoted
    • with and , then
    • Odds of event E :
    • Question : Football game between teams A and B,
    • If P(A win)=0.8, then P(B win)=?
    • Odds (A win)=? Odds (B win)=?
  • 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
  • 8. 2.1.3 Bayes ’ formula
    • Smoking ( A ) -> Lung cancer ( B ) ?
    • Randomly divide the subjects into two groups;
    • Invite one group to smoke and forbid another;
    • Follow up year by year to obtain the number of
    • the subjects “with lung cancer” ……
    • Unfortunately, it is morally infeasible. Then how?
  • 9.
    • To find
    • by Bayes’ formula
    • Example 2.1
  • 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
  • 15.
    • Example Five “exactly same” animals were
    • injected by a poison with dose of LD50 (Under
    • such a dose, the P (death) = 50% )
    • Since
    • The possible number of deaths was ;
    • The probability of each animal died from this
    • injected poison is ;
    • Independently repeated n = 5 times ;
    • X followed
  • 16. 2.4 Poisson Distribution
    • Distribution of rare “articles”
    • Special case of Binomial distribution :
    • Big n , small
    • Example Pulse count of radio active isotope( 同位素 ).
    • Large n and 0-1 : Divide the period into n sub-intervals, possible numbers of pulses in a sub-interval = 0 or 1
    • Rare event :
    • Independent
  • 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
  • 19.
    • Note “independent” and “repeat” are important ,
    • without these two, the distribution will not be a
    • Poisson distribution.
    • Example:
    • For an infectious rare disease, the number of patients does not follow a Poisson distribution at all.
    • When the bacterium are clustered in milk, the total number of bacterium does not follow a Poisson distribution either.
  • 20. 2.4.2 Plot of probability function , positive skew; , approximately symmetric
  • 21. Property of Poisson Distribution
    • population mean = population variance = λ
    • Additive property
    • If and
    • independent each other, then
    • If
    • then
  • 22.
    • If ,
    • then 2 X does not follow
    • does not follow
    However
  • 23. Example : Five samples taken from a river , the number of colibacillus ( 大肠杆菌 ) were counted
    • 1-st sample, X 1 ~  (  1 )
    • 2-nd sample X 2 ~  (  2 )
    • …………… . …… .
    • 5th sample X 5 ~  (  5 )
    • If mix these 5 samples, the total number of
    • colibacillus also follows a Poisson distribution
    • X 1 + X 2 +…+ X 5 ~  (  1 +  2 +…+  5 )
    • Application of additive property:
    • In order to enlarge the parameter, and then make the
    • distribution symmetric, we may pool the small units such that enlarge the observed unit.
  • 24. 2.5 Normal Distribution
    • In practice, The shape of frequency histograms of many
    • continuous random variables looks like this:
    • taller around center, shorter on two sides and symmetric.
  • 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
  • 27. 2.5.2 Area under the normal probability density curve
    • A table for standard normal distribution is usually
    • attached in most textbooks of statistics. (P. 479)
    • ---- Given z , to find out
    z 0
  • 28.
    • The area within
    Corresponding to 1.96, the area of one tail is 0.025, the area of two tails is 0.025  2 = 0.05
  • 29.
    • The area within
    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
  • 30.  
  • 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,
  • 33. 2.5.3 Determination of a reference range
    • Reference range or normal range : The range of most
    • “ healthy people”. “Most” : 95% or 99%
    • “ Healthy people”: should be well defined
    • Determined by a large sample
    • 1. If the variable follows a normal distribution
    • then covers 95% of “healthy people”.
    • However, usually are unknown! They may
    • replaced by (It is why a large sample needed)
    • Therefore, reference range:
  • 34.
    • 2. If the variable does not follow a normal distribution, then find out the percentile and percentile
    • Therefore, reference range:
  • 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.
  • 36. Caution
    • The 95% reference range just tells that the measures of 95% healthy males are within this range;
    • If someone’s measure is falling in this range, can we claim “ normal” ?
    • If someone’s measure is outside this range. can we claim “ abnormal” ?
    • ---- The reference range could never be a criterion for diagnosis.
  • 37. 2.5.4 Normal approximation of binomial distribution and Poisson distribution
    • When n is large enough, (n  >5, n (1-  ) >5) , the
    • binomial distribution approximates to a
    • normal distribution
    • When  is large enough (   20 ) , the Poisson
    • distribution approximates to a normal
    • distribution
  • 38.  
  • 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
  • 41.
    • Example The p ulse count of radio active isotope
    • in 0.5 hour follows a Poisson distribution .
    • Please estimate the probability that the pulse
    • count measured is greater than 400 .
  • 42. Summary
    • Three distributions :
    • Discrete variable : Binomial distribution
    • Poisson distribution
    • Continuous variable : Normal distribution
    • 1. Binomial distribution
    • Possible values: 0 , 1
    • Probability of positive event in one trial =  ,
    • Probability of negative event in one trial = 1 -  ,
    • Independently repeat n times
    • Total number of positive event
    • 2. Poisson distribution
    • When  or ( 1 -  ) is very small , n very large,
    • binomial distribution approximate to Poisson distribution.
  • 43.
    • 3. Normal distribution ---- very important
    • Many phenomena follow normal distributions;
    • Important basis of statistical theory
    • Two parameters :
    • Mean μ Standard deviation σ
    • Z- transformation
    • Area under the curve of normal distribution
  • 44.
    • 4. Normal approximation
    • When n is large ( both of n  and n (1-  ) >5) ,
    • approximates to
    • When  is large (   20 ) ,
    • approximates to
    5. Web resources http://statpages.org/
  • 45.
    • Thanks
  • 46.  

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