Binomial Random Variables - Instructional Web Server -二项随机变量——教学的Web服务器—.pptVIP

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Binomial Random Variables - Instructional Web Server -二项随机变量——教学的Web服务器—.ppt

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Binomial Random Variables - Instructional Web Server -二项随机变量——教学的Web服务器—

Binomial Random Variables Many experiments are like tossing a coin a fixed number of times and recording the up-face. The two possible outcomes are Heads and Tails. Each outcome has an associated probability. For a fair coin, P(Heads) = P(Tails) = .50 X = # of Heads (or Tails) in n trials constitutes a Binomial Random Variable Binomial Random Variables X is binomial because there are only two possible outcomes. In a coin toss, these outcomes are Heads, Tails. X is random because the outcome is not predictable in advance (in practice vs. in principle) X is a variable: the outcome of interest (e.g., the # of Heads in n tosses) varies from one set of n tosses to the next set of n tosses. Binomial Random Variables The same logic works with experiments similar to a series of coin tosses. Our Binomial Random Variable is the # of times an outcome of interest occurs. Example: a sample of UWO students are asked whether they favor building a new rec center. The BRV would be X, the number of students in favor out of n students in the sample. Necessary Conditions for a BRV The experiment consists of n identical trials There are only 2 possible outcomes, called success (S) failure (F) (labels are arbitrary) P(S) = p. P(F) = q = (1-p). P and q are constant across trials. Trials are independent of each other (because p and q are constant across trials) BRV X = # of successes in n trials. 3 ways to compute a binomial probability It is often of interest to know the probability of obtaining X successes in n trials of a binomial experiment. There are 3 ways to determine that probability. Table look-up Binomial expansion formula Normal approximation to the Binomial BRV – Method #1 Our preferred method of determining a binomial probability is to look it up in the Binomial Probability Table, found in Table II, Appendix A, in the text. Table II contains cumulative probabilities – that is, for any X, the table gives the probability of observing X or fewer successes in n