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Last Time

Last Time. Administrative Matters – Blackboard … Random Variables Abstract concept Probability distribution Function Summarizes probability structure Sum to get any prob. Binomial Distribution. Reading In Textbook. Approximate Reading for Today’s Material:

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Last Time

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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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