{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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 .
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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?
Image of page 4
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) .
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.
  • Spring '09
  • WANG
  • Probability theory, Binomial distribution, Discrete probability distribution, Discrete Probability Distributions, Geometric distribution, Hypergeometric

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern