class: center, middle, inverse, title-slide # Bernoulli Distribution ### Dr. Dogucu --- layout: true <div class="my-header"></div> <div class="my-footer"> Copyright &copy; <a href="https://mdogucu.ics.uci.edu">Dr. Mine Dogucu</a>. <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a></div> --- ## Trial Assume a random process with only two outcomes. e.g. a medical test of a disease with positive or negative result. A trial can be thought of a single medical test given to one person. --- ## Success We are interested in `\(p\)` which represents the proportion of people in the population with this disease. A success of a trial is in this case getting a positive result. -- Consider a disease that is found in 8% of the population. Then the probability of success would be 0.08. -- We denote success with 1 and failure with 0. `\(P(X=1) = 0.08\)` `\(P(X=0) = 1 - 0.08 = 0.92\)` --- ## Bernoulli Distribution If X is a random variable that takes value 1 with probability of success `\(p\)` and 0 with probability `\(1-p\)`, then X follows a Bernoulli distribution. `\(X \sim \text{Bernoulli} (p)\)` Expected Value, `\(\mu = p\)` Variance, `\(\sigma^2 = p(1-p)\)`