P&S L2 C1 S2

Intro Stats: Islamic Approach; Part 2: Probabiility and Statistics

Lecture 2: Binomial Random Variables - Concept 1 Bernoulli Random Variables

S2: Sequences of Bernoullis

One Bernoulli is obtained by a simple random draw from a hypothetical population which has two categories or types. A sequence of Bernoullis is obtained via a simple random sample, or a sequence of independent simple random draws from the same population.

Example: A Population of 100 students has 60 males and 40 females. A simple random draw is made from the population. A Bernoulli Variable B1 is recorded to be 1, if the draw is female, and B1 = 0 if the draw if mail. This real world situation CAN BE MODELED by a hypothetical population with 10 hypothetical members six males and four females. A simple random draw from this hypothetical population provides a MODEL for the real world counterpart, since the proportions of the two types match in the hypothetical and real population.

A sequence of simple random draws, or a simple random sample, creates a sequence of Bernoulli random variables B1, B2, B3, .... Each of the random variables have two outcomes 0 and 1 with the same probabilities 60% and 40% These varaibles ae also INDEPENDENT, each random draw is completely independent of the others, so that one has no affect on the other. The standard abbreviation for this setup, which is basic and fundamental to all of the theory of probabilty and statistics, is called an IID sequence. The first I stands for independent, and ID stands for IDENTICALLY DISTRIBUTED, meaning that all variables have the same distribution.