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This article explains the abstract concept of the binomial distribution and its probability distribution function. It discusses the parameters, setting, and examples of the binomial distribution, as well as the binomial probability distribution function. The article also covers sampling with and without replacement, independence, and calculations for desired probabilities.

josephadam
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  1. Last Time • Administrative Matters – Blackboard … • Random Variables • Abstract concept • Probability distribution Function • Summarizes probability structure • Sum to get any prob. • Binomial Distribution

  2. Reading In Textbook Approximate Reading for Today’s Material: Pages 311-317, 327-331, 372-375 Approximate Reading for Next Class: Pages 377-381, 385-391, 488-491

  3. Binomial Distribution Setting: n independent trials of an experiment with outcomes “Success” and “Failure”, with P{S} = p.

  4. Binomial Distribution Setting: n independent trials of an experiment with outcomes “Success” and “Failure”, with P{S} = p. Say X = #S’s has a “Binomial(n,p) distribution”, and write “X ~ Bi(n,p)”

  5. Binomial Distribution Setting: n independent trials of an experiment with outcomes “Success” and “Failure”, with P{S} = p. Say X = #S’s has a “Binomial(n,p) distribution”, and write “X ~ Bi(n,p)” • Called “parameters” (really a family of distrib’ns, indexed by n & p)

  6. Binomial Distribution E.g. Sampling with replacement • “Experiment” is “draw a sample member” • “S” is “vote for Candidate A” • “p” is proportion in population for A (note unknown, and goal of poll) • Independent? (since with replacement)

  7. Binomial Distribution E.g. Sampling with replacement • “Experiment” is “draw a sample member” • “S” is “vote for Candidate A” • “p” is proportion in population for A (note unknown, and goal of poll) • Independent? (since with replacement) X = #(for A) has a Binomial(n,p) dist’n

  8. Binomial Distribution E.g. Sampling without replacement • Draws are dependent Result of 1st draw changes probs of 2nd draw • P(S) on 2nd draw is no longer p (again depends on 1st draw) X = #(for A) is NOT Binomial

  9. Binomial Distribution E.g. Sampling without replacement • Draws are dependent Result of 1st draw changes probs of 2nd draw • P(S) on 2nd draw is no longer p (again depends on 1st draw) X = #(for A) is NOT Binomial (although approximately true for large pop’n)

  10. Binomial Distribution Models much more than political polls: E.g. Coin tossing (recall saw “independence” was good) E.g. Shooting free throws (in basketball) • Is p always the same? • Really independent? (turns out to be OK)

  11. Binomial Prob. Dist’n Func. • Summarize all prob’s for X ~ Bi(n,p)

  12. Binomial Prob. Dist’n Func. • Summarize all prob’s for X ~ Bi(n,p) • By function:

  13. Binomial Prob. Dist’n Func. • Summarize all prob’s for X ~ Bi(n,p) • By function: Recall: • Sum over this for any prob. about X

  14. Binomial Prob. Dist’n Func. • Summarize all prob’s for X ~ Bi(n,p) • By function: Recall: • Sum over this for any prob. about X • Avoids doing complicated calculation each time want a prob.

  15. Binomial Prob. Dist’n Func. Repeat “experiment” (S or F) n times

  16. Binomial Prob. Dist’n Func. Repeat “experiment” (S or F) n times • Outcomes “Success” or “Failure”

  17. Binomial Prob. Dist’n Func. Repeat “experiment” (S or F) n times • Outcomes “Success” or “Failure” • Independent repetitions • Let X = # of S’s (count S’s)

  18. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s

  19. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = Desired probability distribution function

  20. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = Depends on particular draws, So expand in those terms, and use Big Rules of Probability

  21. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] • For “S on 1st draw”, “S on x-th draw”, …

  22. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] • For “S on 1st draw”, “S on x-th draw”, … • One possible ordering of S,…,S,F,…,F where: x of these n-x of these

  23. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] • For “S on 1st draw”, “S on x-th draw”, … • One possible ordering of S,…,S,F,…,F • This includes all other orderings (very many, but we can think of them)

  24. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] Next decompose with and – or – not Rules of Probability

  25. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = = P[(S1&…&Sx&Fx+1&…&Fn)] + … • Disjoint OR rule [“or”  add]

  26. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = = P[(S1&…&Sx&Fx+1&…&Fn)] + … • Disjoint OR rule [“or”  add] (recall “no overlap”)

  27. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = = P[(S1&…&Sx&Fx+1&…&Fn)] + … = P(S1)…P(Sx)P(Fx+1)…P(Fn) + … • Independent AND rule [“and”  mult.]

  28. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = = P[(S1&…&Sx&Fx+1&…&Fn)] + … = P(S1)…P(Sx)P(Fx+1)…P(Fn) + … = since p = P[S] since (1-p) = P[F]

  29. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = = P[(S1&…&Sx&Fx+1&…&Fn)] + … = P(S1)…P(Sx)P(Fx+1)…P(Fn) + … = since x = #S’s since (n-x) = #F’s

  30. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = P[(S1&…&Sx&Fx+1&…&Fn) or …] = = P[(S1&…&Sx&Fx+1&…&Fn)] + … = P(S1)…P(Sx)P(Fx+1)…P(Fn) + … = = #(terms) since all of these are the same, just count

  31. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = #(terms) # ways to order S …S F …F

  32. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = #(terms) # ways to order S …S F …F Approach: have “n slots”

  33. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = #(terms) # ways to order S …S F …F Approach: have “n slots” “choose x of them to in which to put S”

  34. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = #(terms) # ways to order S …S F …F Approach: have “n slots” “choose x of them to in which to put S” thus have #(terms) =

  35. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = #(terms) = general formula that works for all n, p, x

  36. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s P[X = x] = #(terms) = = Binomial Probability Distribution Function (for any n and p)

  37. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s More complete representation

  38. Binomial Prob. Dist’n Func. Repeat (S or F) n times (ind.), let X = # of S’s More complete representation But generally assume is understood, & write

  39. Binomial Prob. Dist’n Func. Application of: For X ~ Bi(n,p) • Compute any probability for X • By summing over appropriate values

  40. Application of Bi. Pro. Dist. Fun. Application of: E.g.: A system fails if any 3 of 5 independent components fail

  41. Application of Bi. Pro. Dist. Fun. Application of: E.g.: A system fails if any 3 of 5 independent components fail • Common setup in Reliability Theory

  42. Application of Bi. Pro. Dist. Fun. Application of: E.g.: A system fails if any 3 of 5 independent components fail • Common setup in Reliability Theory • Used when things “really need to work” • E.g. aircraft components

  43. Application of Bi. Pro. Dist. Fun. Application of: E.g.: A system fails if any 3 of 5 independent components fail If each component works 99% of time,

  44. Application of Bi. Pro. Dist. Fun. Application of: E.g.: A system fails if any 3 of 5 independent components fail If each component works 99% of time, how likely is the system to break down?

  45. Application of Bi. Pro. Dist. Fun. Application of: E.g.: Sys. F if 3 of 5 F, each works 99% time, how likely is the system to break down?

  46. Application of Bi. Pro. Dist. Fun. Application of: E.g.: Sys. F if 3 of 5 F, each works 99% time, how likely is the system to break down? Let X = #F’s

  47. Application of Bi. Pro. Dist. Fun. Application of: E.g.: Sys. F if 3 of 5 F, each works 99% time, how likely is the system to break down? Let X = #F’s, model X ~ Bi(5,0.01)

  48. Application of Bi. Pro. Dist. Fun. Application of: E.g.: Sys. F if 3 of 5 F, each works 99% time, how likely is the system to break down? Let X = #F’s, model X ~ Bi(5,0.01) • Recall n = # of trials (repeats of experim’t)

  49. Application of Bi. Pro. Dist. Fun. Application of: E.g.: Sys. F if 3 of 5 F, each works 99% time, how likely is the system to break down? Let X = #F’s, model X ~ Bi(5,0.01) • Components assumed independent

  50. Application of Bi. Pro. Dist. Fun. Application of: E.g.: Sys. F if 3 of 5 F, each works 99% time, how likely is the system to break down? Let X = #F’s, model X ~ Bi(5,0.01) • Recall p = P(“S”), on each trial (works 99%, so fails 1%)

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