lecture7 - Handout 4 Binomial Distribution Reading...

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Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and per- centiles for those type of variables. Throughout this handout we will discuss discrete random variables - more specifically, binomial random variables. Recall that discrete random variables can take only one of a countable list of distinct values. The Binomial Random Variable Certain conditions must be met for a variable to be considered a binomial random variable, but the basic idea is that a binomial random variable is a count of how many times an event occurs (or does not occur) in a particular number of independent observations or trials that make up a random circumstance. One of the more basic examples of a binomial random variable is the number of heads observed in four tosses of a fair coin. We define a binomial random variable as X = number of successes in the n trials of a binomial process (e.g. X = the number of heads in four tosses of a fair coin). A binomial process is defined by the following conditions: 1. There are n specified ‘trials’. 2. Each observation results in one of two possible outcomes, called ‘success’ and ‘failure’. 3. The probability of a ‘success’ remains the same from one trial to the next, and this probability is denoted by p . The probability of a ‘failure’ is q = 1 - p for every trial. 4. The outcomes are independent from one trial to the next Sometimes there may be more than two possible simple events for each trial (think of rolling a die there are six possible outcomes), but the random variable counts how many times a particular subset of the pos- sibilities occurs (the up-face on the die is an even value). Surveyed responses can also produce a binomial random variable when we count how many individuals in the sample have a particular trait or opinion.

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