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UNR, MATH/STAT 352, Spring 2007

Probability and Statistics. MATH/STAT 352. Spring 2007. Lecture 14: Binomial distribution Poisson distribution Normal distribution. (slides only contain intro). UNR, MATH/STAT 352, Spring 2007. Binomial distribution. number of successes in n Bernoulli trials with probability p of success.

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UNR, MATH/STAT 352, Spring 2007

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  1. Probability and Statistics MATH/STAT 352 Spring 2007 Lecture 14: Binomial distribution Poisson distribution Normal distribution (slides only contain intro) UNR, MATH/STAT 352, Spring 2007

  2. Binomial distribution number of successes in n Bernoulli trials with probability p of success Binomial(n,p) UNR, MATH/STAT 352, Spring 2007

  3. Binomial distribution Binomial(1,0.5) Number of successes within 1 symmetric Bernoulli trial can only be 0 or 1. These possibilities have equal chances. UNR, MATH/STAT 352, Spring 2007

  4. A fair game should result in a tie. Then why do people play fair games? Binomial distribution Binomial(2,0.5) Number of successes within 2 symmetric Bernoulli trials can only be 0, 1 or 2. The possibility to have exactly 1 success is larger than that of having 0 or 2. UNR, MATH/STAT 352, Spring 2007

  5. Binomial distribution Binomial(2,0.1) Most likely there will be no successes UNR, MATH/STAT 352, Spring 2007

  6. Binomial distribution Binomial(2,0.9) Most likely there will be only successes UNR, MATH/STAT 352, Spring 2007

  7. Binomial distribution Binomial(15,0.1) Unimodal (mode = 1) Right-skewed (E = np = 1.5) Concentrated around 1.5 UNR, MATH/STAT 352, Spring 2007

  8. Binomial distribution Binomial(19,0.1) Unimodal (mode = 1-2) Right-skewed Concentrated around 1.5 UNR, MATH/STAT 352, Spring 2007

  9. Binomial distribution Binomial(100,0.1) Unimodal (mode = 10) Symmetric? (E = np = 10) Concentrated around 10 P(10+3) < P(10-3) UNR, MATH/STAT 352, Spring 2007

  10. Binomial distribution Binomial(100,0.9) Unimodal (mode = 90) Symmetric? (E = np = 90) Concentrated around 90 UNR, MATH/STAT 352, Spring 2007

  11. Binomial distribution Bin(100,0.1) Bin(100,0.9) Only a small fraction of possible outcomes has not negligible P (i.e. only small part can be seen in experiment) P is very small (not 0!) here UNR, MATH/STAT 352, Spring 2007

  12. Binomial distribution Binomial(6,0.5) Here all possible outcomes have reasonable probabilities UNR, MATH/STAT 352, Spring 2007

  13. Poisson distribution l number of rare events with average occurrence rate Poisson(l) UNR, MATH/STAT 352, Spring 2007

  14. Binomial(1000,.001] I’ve seen this already! Poisson distribution Poisson(1) UNR, MATH/STAT 352, Spring 2007

  15. Poisson vs. Binomial Poisson is a limit of Binomial, or Binomial can be approximated by Poisson If + + then Binomial(n,p)  Poisson(l) UNR, MATH/STAT 352, Spring 2007

  16. Poisson? Binomial? Normal? Poisson(30) Binomial(1000,.03) N(30,sqrt(30)) Poisson Binomial Normal UNR, MATH/STAT 352, Spring 2007

  17. Poisson? Binomial? Normal? Rule of thumb: If n is large (n > 100), p is small(p < 0.05), and both np and n(1-p) are not small (say >10) then B(n,p)~P(np)~N(np, np(1-p)) UNR, MATH/STAT 352, Spring 2007

  18. Normal distribution UNR, MATH/STAT 352, Spring 2007

  19. Distribution UNR, MATH/STAT 352, Spring 2007

  20. True or false? If a fair game is played long enough, the probability of zero payoff or tie (# wins = # losses) becomes close to 1. UNR, MATH/STAT 352, Spring 2007

  21. Net payoff in N fair games UNR, MATH/STAT 352, Spring 2007

  22. Net payoff in N fair games UNR, MATH/STAT 352, Spring 2007

  23. Net payoff in N fair games UNR, MATH/STAT 352, Spring 2007

  24. Net payoff in N fair games UNR, MATH/STAT 352, Spring 2007

  25. Net payoff in N fair games UNR, MATH/STAT 352, Spring 2007

  26. Net payoff in N fair games UNR, MATH/STAT 352, Spring 2007

  27. Net payoff in N fair games UNR, MATH/STAT 352, Spring 2007

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