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Binomial Distribution

Binomial Distribution. And general discrete probability distributions. Random Variable. A random variable assigns a number to a chance outcome or chance event. The definition of the random variable is denoted by uppercase letters at the end of alphabet, such as W, X, Y, Z.

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Binomial Distribution

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  1. Binomial Distribution And general discrete probability distributions...

  2. Random Variable • A random variable assigns a number to a chance outcome or chance event. • The definition of the random variable is denoted by uppercase letters at the end of alphabet, such as W, X, Y, Z. • The possible values of the random variable are denoted by corresponding lowercase letters w, x, y, z.

  3. Examples: Discrete random variables • W = number of beers randomly selected student drank last night, w = 0, 1, 2, … • X = number of aspirin randomly selected student took this morning, x = 0, 1, 2, ... • Y = number of children in a family with children, y = 1, 2, 3, ...

  4. Discrete Probability Distribution • A discrete probability distribution specifies: • the possible values of the random variable, and • the probability that each outcome will occur

  5. Example: Discrete probability distribution • Let X = number of natural brothers PSU students have. X = 0, 1, 2, ... • P(X=0) = 0.41 • P(X=1) = 0.45 • P(X=2) = 0.11 • P(X=3) = 0.03

  6. Example: Graphically …

  7. Example: Discrete probability distribution • Let X = the number selected when a student picks a number between 0 and 9. • If students pick number randomly, then the probability of picking any number is 0.10. • That is, P(X = 0) = … = P(X = 9) = 0.10

  8. Example: Binomial random variable • Student told 3 statements -- 2 true, 1 false. • Student tries to identify false statement. • Student does this 3 different times. • Let X = the number of times student correctly identifies the false statement. Then, X = 0, 1, 2 or 3.

  9. A special kind of discrete random variable having the following four characteristics: Binomial Random Variable • n identical “trials”: student tries to guess 3 times • 2 possible outcomes denoted “success” or “failure”: student picks either false statement or true statement. • independent trials: a student’s success or failure on one try doesn’t affect success or failure on another try • p = P(“success”) is same for each trial: if just guessing, a student has probability of 1/3 of picking false statement.

  10. Is X binomial? Probability student smokes pot regularly is 0.25. College administrator surveys students until finds one who smokes pot. Let X = number of students surveyed.

  11. Is X binomial? Unknown to quality control inspector, crate of 50 light bulbs contain 3 defective bulbs. QC inspector randomly selects 5 bulbs “without replacement”. Let X = number of defective bulbs in inspector’s sample.

  12. Is X binomial? Unknown to us, the probability an American thinks Clinton should have been removed from office is 0.29. Gallup poll surveys 960 Americans. Let X = number of Americans in sample who think Clinton should have been removed from office.

  13. Is X binomial? Students pick one number between 0 and 9. Let X = number of students who pick the number “7”

  14. Example: Binomial r.v. Let 3 students pick. Let Y = #7 and N = not #7

  15. Binomial Probability Distribution P(X = x) = (# of ways x occurs) × px × (1-p)n-x = n!/[x!(n-x)!] × px × (1-p)n-x Where “n-factorial” is defined as n!= n (n-1) (n-2) … 1 and 0! = 1

  16. Examples: n! 5! = 5 × 4 × 3 × 2 × 1 = 120 4! = 4 × 3 × 2 × 1 = 24 3! = 3 × 2 × 1 = 6 2! = 2 × 1 = 2 1! = 1

  17. Example: Binomial Formula P(X = x) =n!/[x!(n-x)!] × px × (1-p)n-x Guessing game. Let n = 3 and p = 0.33. Then: P(X = 0) = 3!/[0!(3-0)!] × 0.330 × (0.67)3-0 = 6/(1×6) × 1 × 0.673 = 0.30 P(X = 1) = 3!/[1!(3-1)!] × 0.331 × (0.67)3-1 = 3 × 0.33 × 0.672 = 0.44

  18. Example (continued) P(X = 2) = 3!/[2!(3-2)!] × 0.332 × (0.67)3-2 = 3 × 0.1089 × 0.67 = 0.22 P(X = 3) = 3!/[3!(3-3)!] × 0.333 × (0.67)3-3 = 1 × 0.037 × 1 = 0.04 Note: 0.30 + 0.44 + 0.22 + 0.04 = 1 

  19. Using binomial probabilities to draw a conclusion If students did just randomly guess which statement was false, we’d expect the random variable X to follow a binomial distribution? Can we conclude that students did not guess the false statements randomly?

  20. Using binomial probabilities to draw a conclusion If students do indeed pick a number between 0 and 9 randomly, how likelyis it that we would observe the sample we did? Can we conclude that students do not pick numbers randomly?

  21. Using binomial probabilities to draw a conclusion Could the space shuttle Challenger disaster of January 28, 1986 have been better predicted? And therefore prevented?

  22. Moral • Probability calculations are used daily to draw conclusions and make important decisions. • Calculated probabilities are accurate only if the assumptions made are indeed correct. • Always check to see if your assumptions are reasonable.

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