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Understanding Binomial, Poisson, and Normal Distributions

Explore the characteristics and applications of Binomial, Poisson, and Normal distributions in probability and statistics. Understand their modes, skewness, and concentration around specific values. Learn the relation between Poisson and Binomial.

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Understanding Binomial, Poisson, and Normal Distributions

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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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