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- 1. Chapter 8<br />The Binomial and Geometric Distributions<br />
- 2. 8.1 the Binomial Distribution<br />
- 3. Definitions<br /> A Binomial Setting is an scenario where each of the following are true<br />Each observation is either a ‘success’ or a failure’ (2 outcomes)<br />There are a fixed number of observations (n)<br />Observations are independent<br />The probability of success is the same for each observation (p)<br />
- 4. Definitions<br />The Binomial Distribution<br />X = # of successes from a binomial setting<br />Abbreviated with “B(n, p)”where n = number of trails, p = prob. of success<br />
- 5. Definitions<br />The Binomial Distribution Examples<br />“X = The number of heads when 5 coins are flipped”B(5, 0.5)<br />“X = The number of heart cards drawn from a standard deck with replacement after 3 tries.”B(3, ¼)<br />“X = The number of working computer chips in a set of 8 chips if the manufacturer has 0.1% defects”B(8, 0.999)<br />
- 6. Sampling Distribution of a Count<br />Suppose an SRS of size n is drawn from a population with a proportion p for success. <br />Unless there is replacement, this isn’t binomial. Why? <br />If the population is much greater than the sample size, then the sample has an approximate binomial distribution B(n, p)<br />This helps use the binomial distribution in many cases that aren’t exactly a binomial setting.<br />
- 7. Formulas for Binomial Settings<br />Using a tree diagram, it is possible to formulate the binomial distribution.<br />Example X = number of heads when a loaded coin is flipped 3 times (P(heads = 0.3)<br />
- 8. Formulas for Binomial Settings<br />Setting is B(3, .3)<br />Sample Space-<br />HHH, HHT, HTH, THH, HTT, THT, TTH, TTT<br />ProbabilitiesP(X = 3) = 0.027, <br />P(X = 2) = 3·(.3)2(.7) = 0.189, P(X = 1) = 3·(.3)(.7)2=0.441, <br />P(X = 0) = (.7)3=0.343<br />Unfortunately, this works best only for simple settings <br />
- 9. Formulas for Binomial Settings<br />Binomial coefficient<br />This is a part of the formula for the binomial distribution.<br />n = # trails, k = # of success<br />You may have seen this formula before!<br />
- 10. Formulas for Binomial Settings<br />Binomial coefficient<br />Your calculator can find this coefficient for you<br />[math] , “PRB,” “nCr”<br /> example <br />
- 11. Formulas for Binomial Settings<br />Binomial Probability<br />If we have a binomial setting B(n, p) and we want to know P(X = k)<br />
- 12. Formulas for Binomial Settings<br />Binomial Probability<br />Actually your calculator is very efficient in these caculations.The binomial magic is [2nd] [vars] (dist), “binompdf(“ <br />Again assuming B(n. p)<br />
- 13. Some Alphabet Soup<br />“pdf” means “probability distribution function,” which is exactly what we are doing!<br />“binompdf” is the “binomial probability function”<br />This is a discrete probability distribution<br />“cdf” means “cumulative distribution function”<br />This will add together a number of successes. <br />
- 14. Cumulative Binomial Distribution<br />For the question, “if B(n, p), what is the probability k or less successes?”<br />B(5, 0.33)<br /> P(X < 2) = P(X=0) + P(X=1) + P(X=2)<br /> = binomcdf(5, 0.33, 2)<br />Pay attention to where the “equals” goes!<br />Binomcdf(n,p, k) is always P(X < k)<br />
- 15. Cumulative Binomial Distribution<br />If B(6, 0.25), what is the probability of less than 4 successes?<br />B(6, 0.25)<br />P(X < 4) = P(X<3) (always rewrite as “<“)<br /> = binomcdf(6, 0.25, 3)<br /> =0.9624<br />Pay attention to where the “equals” goes!<br />binomcdf(n, p, k) is always P(X < k)<br />
- 16. Cumulative Binomial Distribution<br />For cases involving ‘X>k,’ or ‘X>k’ use the property of complimentary sets<br />Pay attention to where the “=” goes!<br />In B(100, 0.95), what is the probability of more than 90 successes?<br />B(100, 0.95)<br />P(X > 90)= 1- P(X<90) <br /> = 1 - binomcdf(100, 0.95, 90)<br /> =0.9718<br />
- 17. Mean and Standard Deviation<br /> For a binomial distribution B(n, p) the following formulas hold:<br />Remember that these are only for a binomial distribution<br /> We should also note that can be thought of as the “expected value”<br />
- 18. Normal Approximation<br />You should have noticed by now that the Binomial distribution produces a single peak distribution<br />If p is within a certain set of numbers, the distribution is relatively symmetric.<br />Because we like to use the Normal distribution, we have conditions under which the binomial distribution is approximately Normal<br />
- 19. Normal Approximation<br />A binomial distribution is approximately Normal N(np, (npq)) when both np>10 and nq> 10.<br />When using the Normal dist to approximate, be sure to: <br />state “Distribution is approximately Normal: N(np, (npq))”<br />Show that the two conditions above are met<br />Remember that this is an approximation, but it is most often good enough<br />
- 20. 8.2 The Geometric Distribution<br />
- 21. The Geometric Setting<br />The geometric setting is almost like the binomial setting with one major difference:<br />Instead of asking “how many successes,” we ask, “when is the first success?”<br />
- 22. The Geometric Setting<br />Observations are either “success” or “failure”<br />The observations are independent<br />The probability of success is the same for each observation<br />The variable of interest is the number of trails until the first success<br />
- 23. Geometric Distribution<br />If a random variable X satisfies the geometric setting, then we call the distribution of X a geometric distribution <br />FormulaP(X = k) = q(k-1)p<br />notice that this is (k-1) failures and one success<br />
- 24. Geometric Distribution on the TI<br />Like the Binomial Distribution, the Geometric Distribution is found at[2nd] [var] (dist)<br />for G(p)P(X=k) = q(k-1)p = geompdf(p,k)<br />
- 25. Geometric cdf<br />The probability that the first success is within the first k trails can be given with:<br />G(p)<br />P(X < k) = P(X=0) + P(X=1)+ … +P(X=k) =geomcdf(p, k)<br />
- 26. Geometric cdf<br />What is the probability that the first “six” is rolled before four throws of a die?<br />Pay attention to the “=“ sign<br />G(1/6)<br />P(X < 4) = P(X < 3)<br /> = geomcdf(1/6, 3)<br /> = 0.4213<br />
- 27. Geometric cdf<br />Use the compliment properties to find P(X>k)<br />G(p)<br />P(X > k) = 1 – P(x < k)<br />And… pay attention to the ‘equal’ sign!!<br />
- 28. Geometric cdf<br />What is the probability that we roll a number less than 3 after 5 throws?<br />G(2/6) (this corresponds to ‘1’ and ‘2’)<br />P(X > 5) = 1 – P(X < 5)<br /> = 1 – geomcdf(2/6, 5)<br /> = 0.1317<br />
- 29. Geometric cdf<br />Alternatively, the probability that it takes more than k trails to see the first success can be given by:<br />P(X > k) = (1 – p)k<br />
- 30. Mean and Standard Deviation<br />Mean of a geometric distribution is given by:<br /> = 1/p<br />This is the expected value for the first success<br />“on average, the first success occurs on the 1/p trail”<br />
- 31. Mean and Standard Deviation<br />Standard Deviation:<br />This is not a Normal distribution, so don’t try to calculate z-scores and Normalcdf!<br />

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