Economic Statistics

1. “What is the life expectancy for Caucasian males that have survived 54 years?” The answer will almost certainly be larger than 72. I want an estimate “conditioned on” the fact that I have already lived 54 years. Life is dynamic. Things are constantly changing. However, too often decisions are based on static information. Bayes theorem takes advantage of dynamic information to give a better, more correct answer.
Bayes Theorem is a mathematical representation that helps one to calculate conditional probability. It relates inverse representation of the probabilities concerning two events. This theorem is named after the British Mathematician Thomas Bayes.
It is represented by
P(A, B)= P(A B) P(B) or P (A, B)=P(B A)P(A)
So
P(A|B)P(B) = P(B|A)P(A)
The Law of Total Probability:
P(B) = P(B/A).P(A) + P(B/A′) . P(A′)
Total probability and multiplication rule:
P(A/B) = P(B/A).P(A) (multiplication rule) P(B/A).P(A) + P(B/A′) . P(A′) (Law of Total Prob.)
Where:
P(A): probability of occurrence of event A (marginal)
P(B): probability of occurrence of event B (marginal)
P(A′) = probability that A does not occur
P(B/A′) = probability that event B occurs given that A has not occurred already.
P(A,B): Probability of simultaneous occurrence of events A and B (joint)
P(A|B): Probability of occurrence of A given that B has occurred (conditional)
P(B|A): Probability of occurrence of B given that A has occurred (conditional).

2. A continuous random variable is a random variable where the data can take infinitely many values. For example, a random variable measuring the time taken for something to be done is continuous since there are an infinite number of possible times that can be taken.

For any continuous random variable with probability density function f(x), we have that: