{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

week5class - Probability Models A Bernoulli Trial A random...

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Probability Models A Bernoulli Trial A random variable X is a Bernoulli random variable (or Bernoulli trial) if the following conditions are met: X has only two possible outcomes (called success and failure) The probability of success, p , is constant for each trial The trials are independent. Examples of Bernoulli random variables: The Geometric Model Suppose that repeated Bernoulli Trials each having probability of success p are performed until a success occurs. If we denote with X the number of trials required until a first success occurs then X is a Geometric random variable and The expected number of trials required until a success is observed (expected value) for a Geometric random variable is
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
Example: Let’s go back in time to the spring of 2000. How many girls is Adam going to have to ask until 1 of them agrees to go to the senior prom with him? The first answer that comes to mind is 0 since Adam is so cool that all the girls would be asking him to the prom. Let’s assume though, that Adam’s true success rate with ladies is 20% and that this percent is constant for all girls Adam asks. What’s the probability that The fifth girl Adam asks is the first girl to say “yes”? (or say “fine, whatever.”) Adam gets a “yes” in three attempts or less? What’s the expected number of girls Adam will have to ask until one will go to the prom with him? If Adam can take multiple girls to the prom, what’s the probability that the first girl to say “no” to him is the fourth girl he asks?
Image of page 2
Binomial Distributions Binomial distributions are models for some categorical variables, typically representing the number of successes in a series of n trials. Observations must meet the following requirements in order for the overall process to be deemed binomial: The total number of observations, n, must be fixed in advance. Each observation has only two possible outcomes, “success” ( S ) and “failure” ( F ). All n observations must have the same probability of success, p . The observations are all independent of each other, meaning the outcome of each observation is not affected by the outcomes of other observations. Example: Randomly draw n balls with replacement from an urn containing 10 red balls and 20 black balls. Use S to denote the outcome of drawing a red ball and F to denote the outcome of a yellow ball (here we are defining a red ball to be a success and a yellow ball to be a failure). Question : Would it still be a binomial experiment if the balls were drawn without replacement?
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
Binomial Random Variables & Distributions Binomial Random Variable A binomial random variable counts the number of successes in n trials of the binomial experiment.
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ 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