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6 - Discrete Random Variables (Ch. 4 Daniels, Ch. 6 Gerstman) This is an empirical discrete probability distribution for the X = # of pigs in a litter A discrete probability density function (pdf) is defined as f ( x) P( X x) . The Prob column contains estimated values for the probability function. If this study were repeated with a different sample n = 378 litters these estimates would be different. Binomial Distribution Binomial Distribution and Random Variable A binomial random variable X is defined to the number of “successes” in n independent trials where the P(success) = p is constant. In the definition above notice the following conditions need to be satisfied for a binomial experiment: 1. There is a fixed number of n trials carried out. 2. The outcome of a given trial is either a “success” or “failure”. 3. The probability of success (p) remains constant from trial to trial. P( success ) p and P( failure ) 1 p q 4. The trials are independent, the outcome of a trial is not affected by the outcome of any other trial. Binomial Probability Function n n! P( X x) p x q n x p x q n x , x 0,1,..., n x!(n x)! x the coefficient in front denotes the number of ways to obtain x successes in n trials. Example 1: A drug company claims that 10% of patients taking the drug will experience adverse side effects. To test this claim a researcher administers the drug to a random sample of 20 patients. Let X = the number patients in our sample who experience adverse side effects. a) What is the probability that exactly 4 patients experience side effects? b) What is the probability that 2 patients or less experience side effects? 1 c) What is the probability that 5 patients of less experience side effects? d) What is the probability that 6 or more patients experience side effects? e) Suppose that in our sample 6 patients experience side effects. What do think about the claim made by the drug company on the basis of this result? f) Suppose we gave the drug to 100 patients. If the drug companies claim regarding side effects is true, how many side effects do we expect to observe, i.e. what is the mean or expected value of X? 2 g) What is the standard deviation of the number of side effects? Example 2: Resting Energy Expenditure (REE) for Patient with Cystic Fibrosis A researcher believes that patients with cystic fibrosis (CF) expend greater energy during resting than those without CF. To obtain a fair comparison she matches 13 patients with CF to 13 patients without CF on the basis of age, sex, height, and weight. She then measured there REE for each pair of subjects and compared the results. The following results were obtained: Pair CF Healthy Difference Sign of Difference (C) (H) =C-H (+ and -) 1 1153 996 2 1132 1080 3 1165 1182 4 1460 1452 8 + 5 1634 1162 472 + 6 1493 1619 -126 7 1358 1140 218 + 8 1453 1123 330 + 9 1185 1113 72 + 10 1824 1463 361 + 11 1793 1632 161 + 12 1930 1614 316 + 13 2075 1836 239 + What did we conclude regarding the research hypothesis? 3 4 Example 2: Success of New Medical Treatment The current method used to treat certain form of childhood cancer has a 60% success rate. A new method has been proposed that will hopefully have an even higher success rate. To test the new method a sample n = 20 patients with this form of cancer are treated and we find all but 4 patients experience remission. Can we conclude on the basis of this result the new method has a higher success rate/lower failure rate? 5 Binomial Table Generator in JMP To use the Binomial Table Generator file in JMP you simply need to change the number of trials (n) and the probability of success (p) which is labeled a p in the table below. To change the n and p values right-click at the top of the column and change the number in the formula to your desired values. The table will then automatically update, and give the probabilities in the last four columns. The procedure for changing n and p is outlined on the following page. 6 Procedure: 1) Right-click either the n (number of trials or p (probability of success) columns. 2) Click Formula… from the menu that appears when you right-click. 3) Change the number to 7