2. AP Statistics

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4. Binomial and Geometric Distributions Chapter 8

5. Definitions of a Binomial and Geometric Distributions Definition of a Binomial Distribution –The distribution of the count X of successes in the binomial setting is the Binomial Distribution. –X is B(n,p) n is the number of observations. P is the probability of each of the observations Definition of a Geometric Distribution –The distribution of the count n of trials in the geometric setting is the Geometric Distribution. –X is G(n,p) n is the number of observations. P is the probability of how long the success would take to happen once.

6. Definitions of Settings The Binomial Setting –Each observation falls into one of just two categories, which for convenience are called “successes” and “failures” –There is a fixed number of n observations –All observations are independent. –The probability is the same for each observation. The Geometric Setting –Each observation falls into one of just two categories, which for convenience are called “successes” and “failures” –All observations are independent. –The probability is the same for each observation. –The variable of interest is the number of trials required to obtain the first success.

7. Binomial Distributions A Binomial Distribution has a probability distribution function…or a Binomial PDF. The Binomial PDF is located on a TI calculator under 2nd / DISTR / 0:binompdf. This is used to find the binomial probability of a value X. So, it is used to find an exact match to a number X. A Binomial Distribution has a cumulative distribution function… or a Binomial CDF. The Binomial CDF is located on a TI calculator under 2nd / DISTR / A:binomialcdf. This is used to find the binomial probability of a value X. So, it is used to find the added probabilities up to a number X.

8. Binomial and Geometric PDF and CDF Binomial PDF is written as binomialpdf(n, p, x) –n is trials –p is probability –x is number you are looking for Binomial CDF is written as binomialcdf(n, p, x) –n is trials –p is probability –x is number you looking for up to that number Geometric PDF is written as geometricpdf(n, p, x) –n is number of trials required to reach one success –p is probability –x is number of successes needed Geometric CDF is written as geometriccdf(n, p, x) –n is number of trials required to reach one success –p is probability –x is number of successes needed

9. Important Notes The approximation is based on finding an area under the normal curve that approximates the y-coordinate of the appropriate binomial probability plot. In order to use the normal approximation to answer binomial questions the binomial is discrete and the normal is continuous, the values of interest must sometimes be adjusted. When n is large, it is unnecessary to apply the continuity correction for an interval of values since the error introduced by not applying it is very small.

10. A Few Shortcuts and Hints 1-seq (x, x, 0, 10, 1) L1 2-binompdf (10,.1, L1) L2 will calculate each P(x =…) L 1 L 2 L 3 1..34868 2..38742 3..19371 4..0574 5. … 6. … 3- Turn off any Y= “stuff” 4- [STAT] [PLOT]… histogram X list = L1 Y list = L2 5- Set window X[-.5, 10.5] Y[-.1,.45]