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Lecture Unit 4 Section 4.2

Lecture Unit 4 Section 4.2. Random Variables and Probability Models: Binomial, Geometric and Poisson Distributions. Streamline Treatment of Probability. Sample spaces and events are good starting points for probability

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Lecture Unit 4 Section 4.2

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  1. Lecture Unit 4Section 4.2 Random Variables and Probability Models: Binomial, Geometric and Poisson Distributions

  2. Streamline Treatment of Probability • Sample spaces and events are good starting points for probability • Sample spaces and events become quite cumbersome when applied to real-life business-related processes • Random variables allow us to apply probability, risk and uncertainty to meaningful business-related situations

  3. Lecture Unit 2 Section 4.1 Bring Together Lecture Unit 2, and Section 4.1 • In Lecture Unit 2 we saw that data could be graphically and numerically summarized in terms of midpoints, spreads, outliers, etc. • In Section 4.1 we saw how probabilities could be assigned to outcomes of an experiment. Now we bring them together

  4. First: Two Quick Examples 1. Hardee’s vs. The Colonel

  5. Hardee’s vs The Colonel • Out of 100 taste-testers, 63 preferred Hardee’s fried chicken, 37 preferred KFC • Evidence that Hardee’s is better? A landslide? • What if there is no difference in the chicken? (p=1/2, flip a fair coin) • Is 63 heads out of 100 tosses that unusual?

  6. Example 2.Mothers Identify Newborns

  7. Mothers Identify Newborns • After spending 1 hour with their newborns, blindfolded and nose-covered mothers were asked to choose their child from 3 sleeping babies by feeling the backs of the babies’ hands • 22 of 32 women (69%) selected their own newborn • “far better than 33% one would expect…” • Is it possible the mothers are guessing? • Can we quantify “far better”?

  8. Graphically and Numerically Summarize a Random Experiment • Principal vehicle by which we do this: random variables • A random variable assigns a number to each outcome of an experiment

  9. Random Variables • Definition: A random variable is a numerical-valued function defined on the outcomes of an experiment Random variable S Number line

  10. Examples • S = {HH, TH, HT, TT} the random variable: • x = # of heads in 2 tosses of a coin • Possible values of x = 0, 1, 2

  11. Two Types of Random Variables • Discrete: random variables that have a finite or countably infinite number of possible values • Test: for any given value of the random variable, you can designate the next largest or next smallest value of the random variable

  12. Examples: Discrete rv’s • Number of girls in a 5 child family • Number of customers that use an ATM in a 1-hour period. • Number of tosses of a fair coin that is required until you get 3 heads in a row (note that this discrete random variable has a countably infinite number of possible values: x=3, 4, 5, 6, 7, . . .)

  13. Two types (cont.) • Continuous: a random variable that can take on all possible values in an interval of numbers • Test: given a particular value of the random variable, you cannot designate the next largest or next smallest value

  14. Which is it, Discrete or Continuous? • Discrete random variables “count” • Continuous random variables “measure” (length, width, height, area, volume, distance, time, etc.)

  15. Examples: continuous rv’s • The time it takes to run the 100 yard dash (measure) • The time between arrivals at an ATM machine (measure) • Time spent waiting in line at the “express” checkout at the grocery store (the probability is 1 that the person in front of you is buying a loaf of bread with a third party check drawn on a Hungarian bank) (measure)

  16. Examples: cont. rv’s (cont.) • The length of a precision-engineered magnesium rod (measure) • The area of a silicon wafer for a computer chip coming off a production line (measure)

  17. Classify as discrete or continuous • x=the number of customers who enter a particular bank during the noon hour on a particular day • discrete x={0, 1, 2, 3, …} • x=time (in seconds) required for a teller to serve a bank customer • continuous x>0

  18. Classify (cont.) • x=the distance (in miles) between a randomly selected home in a community and the nearest pharmacy • continuous x>0 • x=the diameter of precision-engineered “5 inch diameter” ball bearings coming off an assembly line • continuous; range could be {4.5<x<5.5}

  19. Classify (cont.) • x=the number of tosses of a fair coin required to observe at least 3 heads in succession • discrete x=3, 4, 5, ...

  20. Data Variables and Data Distributions Data variables are known outcomes.

  21. Data Variables and Data Distributons Data variables are known outcomes. Data distributions tell us what happened. Handout 2.1, P. 10

  22. Random Variables and Probability Distributions Random variables are unknown chance outcomes. Probability distributions tell us what is likely to happen. Data variables are known outcomes. Data distributions tell us what happened.

  23. Profit Scenarios Economic Scenario Profit ($ Millions) Random variables are unknown chance outcomes. Probability distributions tell us what is likely to happen. Great 10 5 Good Handout 4.1, P. 3

  24. Profit Scenarios Economic Scenario Profit ($ Millions) Great 10 5 Good OK 1 Lousy -4

  25. Probability Economic Scenario Profit ($ Millions) Probability The proportion of the time an outcome is expected to happen. Great 10 0.20 5 Good 0.40 OK 1 0.25 Lousy -4 0.15

  26. Probability Distribution Economic Scenario Profit ($ Millions) Probability Shows all possible values of a random variable and the probability associated with each outcome. Great 10 0.20 5 Good 0.40 OK 1 0.25 Lousy -4 0.15

  27. Economic Scenario Profit ($ Millions) Probability X Notation Great 10 0.20 x1 5 Good 0.40 x2 X = the random variable (profits) xi = outcome i x1 = 10 x2 = 5 x3 = 1 x4 = -4 OK 1 0.25 x3 Lousy -4 0.15 x4

  28. Economic Scenario Profit ($ Millions) Probability X Notation Great 10 0.20 x1 Pr(X=x1) 5 Good 0.40 x2 Pr(X=x2) P is the probability p(xi)= Pr(X = xi) is the probability of X being outcome xi p(x1) = Pr(X = 10) = .20 p(x2) = Pr(X = 5) = .40 p(x3) = Pr(X = 1) = .25 p(x4) = Pr(X = -4) = .15 OK 1 0.25 x3 Pr(X=x3) Lousy -4 0.15 x4 Pr(X=x4)

  29. Economic Scenario Profit ($ Millions) Probability X Great 10 0.20 x1 5 Good 0.40 x2 OK 1 0.25 x3 Lousy -4 0.15 x4 What are the chances? What are the chances that profits will be less than $5 million in 2013? P(X < 5) = P(X = 1 or X = -4) = P(X = 1) + P(X = -4) = .25 + .15 = .40

  30. Economic Scenario Profit ($ Millions) Probability X P Great 10 0.20 x1 p(x1) 5 Good 0.40 x2 p(x2) OK 1 0.25 x3 p(x3) Lousy -4 0.15 x4 p(x4) What are the chances? What are the chances that profits will be less than $5 million in 2013 and less than $5 million in 2014? P(X < 5 in 2013 and X < 5 in 2014) = P(X < 5)·P(X < 5) = .40·.40 = .16 P(X < 5) = .40

  31. Economic Scenario Profit ($ Millions) Probability X Great 10 0.20 x1 p(x1) 5 Good 0.40 x2 p(x2) OK 1 0.25 x3 p(x3) Lousy -4 0.15 x4 p(x4) Probability Histogram Probability .40 .40 .35 .35 .30 .30 .25 .25 .20 .20 .15 .15 .10 .10 .05 .05 -4 -2 0 2 4 6 8 10 12 Profit

  32. Economic Scenario Profit ($ Millions) Probability X P Great 10 0.20 x1 p(x1) 5 Good 0.40 x2 p(x2) OK 1 0.25 x3 p(x3) Lousy -4 0.15 x4 p(x4) Probability Histogram Good Probability .40 .40 .35 .35 OK .30 .30 Great .25 .25 Lousy .20 .20 .15 .15 .10 .10 .05 .05 -4 -2 0 2 4 6 8 10 12 Profit

  33. Probability distributions: requirements • Notation: p(x)= Pr(X = x) is the probability that the random variable X has value x • Requirements 1. 0  p(x)  1 for all values xof X 2. all xp(x) = 1

  34. x p(x) 0 .20 1 .90 2 -.10 property 1) violated: p(2) = -.10 x p(x) -2 .3 -1 .3 1 .3 2 .3 property 2) violated: p(x) = 1.2 Example

  35. Example (cont.) x p(x) -1 .25 0 .65 1 .10 OK 1) satisfied: 0  p(x)  1 for all x 2) satisfied: all xp(x) = .25+.65+.10 = 1

  36. Example: light bulbs 20% of light bulbs last at least 800 hrs; you have just purchased 2 light bulbs. X=number of the 2 bulbs that last at least 800 hrs (possible values of x: 0, 1, 2) Find the probability distribution of X • S: bulb lasts at least 800 hrs • F: bulb fails to last 800 hrs • P(S) = .2; P(F) = .8

  37. Example (cont.) S - SS S F - SF S - FS F Possible outcomes P(outcome) x (S,S) (.2)(.2)=.04 2 (S,F) (.2)(.8)=.16 1 (F,S) (.8)(.2)=.16 1 (F,F) (.8)(.8)=.64 0 probability x 0 1 2 distribution of x: p(x) .64 .32 .04 F - FF

  38. 3 child family; X=#of boys M: child is male P(M)=1/2 (0.5121; from .5134) F: child is female P(F)=1/2 (0.4879) Outcomes P(outcome) x MMM (1/2)3=1/8 3 MMF 1/8 2 MFM 1/8 2 FMM 1/8 2 MFF 1/8 1 FMF 1/8 1 FFM 1/8 1 FFF 1/8 0 Example: 3-child family

  39. Probability Distribution of x x 0 1 2 3 p(x) 1/8 3/8 3/8 1/8 Probability of at least 1 boy: P(x  1)= 3/8 + 3/8 +1/8 = 7/8 Probability of no boys or 1 boy: p(0) + p(1)= 1/8 + 3/8 = 4/8 = 1/2

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