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4_Discrete-1

# 4_Discrete-1 - i i 4 Some Discrete Probability...

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4 Some Discrete Probability Distributions Chapter Outline 4.1 Uniform Distributions 4.2 Bernoulli Distributions 4.3 Binomial Distributions 4.4 Geometric Distributions 4.5 Negative Binomial Distributions 4.6 Hypergeometric Distributions 4.7 Poisson Distributions 4.8 Summary 練習題
2 Chapter 4 Some Well-Known Discrete Probability Distributions 4.1 Uniform Distributions Definition 4.1 If the pmf of a random variable X is f X (x) = 1 /n, x = x 1 , · · · , x n 0 , otherwise, we call that X follows a discrete uniform distribution with parameters x 1 , · · · , x n ; X uniform (x 1 , · · · , x n ). Theorem 4.1 The mean and variance of a discrete uniform distribution: If X uniform (x 1 , · · · , x n ), then E (X) = n i = 1 x i /n, V (X) = n i = 1 x 2 i /n − [ n i = 1 x i /n ] 2 . Definition 4.2 If the pmf of a random variable X is f X (x) = 1 /n, x = a, a + c, a + 2 c, · · · , b 0 , otherwise, where n = 1 + b a c , we call that X follows a discrete equal- spaced uniform distribution with parameters a, b, n ; X uniform (a, b, n ). Theorem 4.2 The mean and variance of the discrete equal-spaced uniform distribution: If X uniform (a, b, n ), then E (X) = (a + b)/ 2 , V (X) = c 2 (n 2 1 )/ 12 .

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4.2 Bernoulli Distributions 3 Example 4.1 How do we use a probability model to describe the outcome of flipping a fair die? What are the expected value and variance of the outcome? Example 4.2 The digits of φ : Mathematicians have spent much time and effort in computing φ , and up to year 1999 there have been 51 billion digits found. Have you ever thought about the probability of 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 as a digit of φ ? In other words, how do we use the probability model to describe the digits? 4.2 Bernoulli Distributions Definition 4.3 If the pmf of a random variable X is f X (x) = p, x = 1 1 p, x = 0 0 , otherwise , we call that X follows a Bernoulli distribution with a param- eter p ; X Bernoulli (p) . Theorem 4.3 The mean and variance of the Bernoulli distribution: If X Bernoulli (p) , then E (X) = p, V (X) = p( 1 p) . Example 4.3 How do we use a probability model to describe the ratio of infant boys to infant girls?
4 Chapter 4 Some Well-Known Discrete Probability Distributions 4.3 Binomial Distributions Definition 4.4 If trial has two possible outcomes that can be labeled as suc- cess or failure, this trial is called a Bernoulli trial. Definition 4.5 If a process is consisted of repeated Bernoulli trials, the pro- cess is referred to as a Bernoulli process. Bernoulli process must possess the following properties: 1. The experiment consists of n repeated trials. 2. Each trial results in an outcome that may be classified as a success or a failure. 3. The probability of success, denoted by p , remains con- stant from trial to trial. 4. The repeated trials are independent. Definition 4.6 If the pmf of a random variable X is f X (x) = C n x p x ( 1 p) n x , x = 0 , 1 , 2 , 3 , . . ., n 0 , otherwise, we call that X follows a Binomial distribution with parame- ters n and p ; X binomial (n, p) or X bin (n, p) .

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• Spring '09
• WANG
• Probability theory, Binomial distribution, Discrete probability distribution, Discrete Probability Distributions, Geometric distribution, Hypergeometric

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