L2 C4 S1

Introduction to Statistics: An Islamic Approach -- Part 2: Probability and Statistics

Lecture 2: Bernoulli Sequences, Concept 4 Bernoulli Probabilities

S1: Probability of A Particular Outcome

1: Consider a population Q which has T elements Q1, Q2, ..., QT. Suppose the first S elements have a certain characteristic, which we label {1} while the remaining T-S elements do not have this characteristic, and are labeled {0}. For example, the population consists of students with Roll Numbers from 1,2,..,T and the first S Roll Numbers have been assigned to Male Students, while the T-S Roll numbers S+1, S+2,...,T have been assigned to Female Students. The characteristic of being male is represented by "1" while being female is represented by "0".

2: Next consider a sequence of Random Draws R1, R2, ...,RN from this population. These draws are done independently, and they are done with replacement, so that all of them have the SAME distribution. In this case we call them IID Random Draws -- Independent and Identically Distributed. The OUTCOMES of these random draws (the students chosen in the process) are a Simple Random Sample from the Population.

3: With this sequence of random draws, we can associate a sequence of IID Random Variables which records the characteristic {1,0} associated with the element of the population chosen randomly. Call this sequence of Bernoulli RV's B1, B2, ... , BN. Then B(i) = 1 if the i-th student drawn is a Male, and B(i)=0 if the i-th student drawn is Female. Let p=S/T be the proportion of Male Students in the Population. Then P(B(i)=1)=p and P(B(i)=0)=1-p. These probabilities remain the same for all of the Bernoulli Random Variables in the sequence.

4: Our goal in this slide is to calculate the probability of any particular sequences of outcomes for the Bernoulli Random Variables