The whitehouse problem

[matsarou]

What is the probability that any famous person (like Shaq) can drop by the White House without an appointment?

Problem statement One attempt to get into the White House represents one trial. n: the number of trials y: the number of observed successes p: the probability of success

• We assume that the trials independent.(a key assumption of the binomial distribution is that the trials are independent)

Solution step 1. define hypotheses, they are the alternative hypotheses for p = a range in [0,1] step 2. find the prior densities, (a probability density shows the belief that each hypothesis is true) step 3. gather the data - one attempt with fail or success. step 4. determine the likelihood of the observed data, assuming each hypothesis is true. step 5. compute the posterior densities for each value of p (using the Bayes theorem)

[matsarou]

• In step 2, the priors is a beta distribution defined in [0,1], with two hyper-parameters a0 and b0
• In step 3, we use a binomial pmf probability.(if it is one attempt, we have bernouli). Shaq makes attempts and we say the result is fail => y=0
• In step4, we say the likelihood is the results of the binomial probability in the step 3 L(y=0|p=0.3,n=1)
• In step5, we compute the posterior of p given the observed data. Instead of p, we use the theta, as P(theta|data) because the p or else the hypotheses are infinite in the range [0,1] The posterior is a beta distribution, with two hyper-parameters a and b The current posterior distribution is the input, as prior, in the next round.