P&S L2 C1 S3
Intro Stats Islamic Approach Part 2: Probability and Statistics
Lecture 2: Binomial Random Variables Concept 1 Bernoulli RV
S3 A Flexbile Model
The Bernoulli Model is useful for many real world situations. We illustrate some situations which can be modelled as Bernoulli sequences.
A sequence of coin flips: Each coin flip can be modelled as a random draw from a hypothetical population with two objects: {Heats, Tails}, if the coin is fair: That is if the coin is symmetric physically and the flipping mechanism does not create preference for either side, then the two sides should appear equally often. Independence seems a reasonable assumption, and Identical Distributions will hold if the coin is fair and remains fair throughout the trials
Voter samples. We wish to determine the percentage for votes for candidates X. We draw a simple random sample and ask whether or not they are voting for X. This can be modelled as a Bernoulli RV. We recorde B1=1 if candidate is voting for X and B1=0 if not. REMEMBER to distinguish between the real world population and real world draws from the IDENTICAL hypothetical population with PERFECT random draws.
A MORE COMPLEX SITUATION: a large population of people with malaria, we choose a random sample of 1000 and give them a new drug Q for the treatment of malaria. After a week, all patients are tested to see if they continue to have malaria or if they have been cured. We would like to test what percentage of the patients are cured by this new drug Q.
CREATE a model for this situation. ASSUME that for each patient, the outcome of the trial is 1 if he is cured and 0 if not. ASSUME that there is a probability p of 1 and 1-p of 0 which is the same for all patients. ASSUME that cure probabilities for all patients are independent of each other. UNDER these assumptions there is a match between a BERNOULLI random variable B1 which is 1 with probability p and 0 with probability 1-p. Now all the patients can be modelled as a sequence of Bernoulli trials B1, b2, ... , b1000, Where there are all independent trials and have identical distributions.