STAT 3021 Chapter5.pdf

STAT 3021 Chapter5.pdf - STAT 3021 Spring 2018 Chapter 5...

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STAT 3021 Spring 2018 Chapter 5. Some discrete probability distributions. 5.1 Introduction and motivation In this chapter we will be learning some of the most frequently used discrete probability distributions such as the Bernoulli distribution, the binomial distribution (5.2), the hyper-geometric distribution (5.3), and so on. The basic idea that we will follow is that when necessary conditions are met , we can derive a general formula for the probability mass function of a discrete random variable X , that is P ( X = x ) = f ( x ). We can then use that formula to calculate probabilities concerning X . We will also learn formulas for the mean (expectation), variance, and standard deviation of X . 5.2 Binomial distributions An experiment often consists with two possible outcomes that may be labeled as success or failure . The following is the simplest form of those experiment. Example 1. Suppose that you have a huge Reese’s piece candy machine. It is known that 50% of the candies are orange, 25% are yellow, and the other 25% are brown. You are going to draw one candy. Let a random variable X be 1 if it is yellow (‘success’) and 0 if it is orange or other (‘failure’). Construct the probability distribution of X . Find the mean and variance of X . The random variable X in Example 1 is called “Bernoulli random variable with the probability of success of p = 1 4 ”. It takes 1 for the category of interest (‘success’) and 0 for the other categories (‘failure’). The probability distribution of Bernoulli random variable is called Bernoulli distribution. For X Bernoulli( p ), E ( X ) = p and V ar ( X ) = p (1 - p ) . 1
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STAT 3021 Spring 2018 Bernoulli process occurs when a Bernoulli experiment is performed for several independent times. Binomial distribution Binomial distribution is the most important discrete probability distribution. Before introducing the defini- tion of Binomial distribution, we will revisit the Reese’s candy machine example. Example 2. (Refer to the description in Example 1). This time you are going to draw three candies. Find the probability that you draw exactly two yellows. In Example 2, the random variable X , the number of yellows in 3 randomly chosen candies, is the number of ‘success’ in the fixed number of 3 Bernoulli trials. 2
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STAT 3021 Spring 2018 Probability mass function of Binomial distribution Note : The constants n and p are called the parameters of the binomial distribution. Example 3. (Refer to Example 2). Construct the probability distribution of X = number of yellows in 3 randomly selected candies. Find the mean and the variance. Since the probability distribution of any binomial random variable depends only on the values of p and n , it would seem reasonable to assume that the mean and variance of a binomial random variable also depend on those parameters.
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