170717P - PRINCIPLES OF EPIDEMIOLOGY HEALTH RESEARCH COURSE: THE PROBABILITY THEORY AS A BASIS FOR DISCRETE RANDOM VARIABLES (RVS)

Presentation at a Course on Principles of Epidemiology, Health Research Faculty of Medicine, King Fahad Medical City, October 11-12, 2017 by Professor Omar Hasan Kasule Sr. MB ChB (MUK). MPH (Harvard), DrPH (Harvard) Chairman of the Institutional Review Board / Research Ethics Committee at King Fahad Medical City, Riyadh.

LECTURE 2: THE PROBABILITY THEORY AS A BASIS FOR DISCRETE RANDOM VARIABLES (RVS)

PROBABILITY AS A CONCEPT:

• The bulk of statistical theory is probability theory since modern inferential statistics depends on probability theory.

• Probability is the modeling of chance random events and a measure of the likelihood of their occurrence.

PROBABILITY AS a CONCEPT, Con’t...:

• Probability is commonly defined as the relative frequency of an event on repeated trials under the same conditions.

• Special mathematical techniques called arrangements, permutations, and combinations, can enable us to calculate the probability space theoretically without having to carry out the trials.

CLASSIFICATION OF PROBABILITY: 

• Probability can be subjective (based on personal feelings or intuition) or objective (based on real data or experience). Objective probability can be measured or computed.

• Prior probability is knowable or calculable without experimentation. The posterior probability is calculable from the results of experimentation.

• Bayesian probability combines prior probability (objective, subjective, or a belief) with new data (from experimentation) to reach a conclusion called posterior probability.

TYPES OF PROBABILITY EVENTS:

• On the scale of exclusion, events are classified as mutually exclusive or non-mutually exclusive. Mutually exclusive events are those that cannot occur together like being dead and being alive.

• On the scale of independence, events are classified as independent or dependent. Under independence, the occurrence of one event is not affected by the occurrence or non-occurrence of another. Independent events can occur at the same instant or subsequently. Some independent events are equally likely while others are not.

• On the scale of exhaustion, two events A and B are said to be exhaustive if between them they occupy all the probability space.

SET THEORY:

• Intersection A n B

 

• Union A u B

 

QUALITATIVE RANDOM VARIABLES:

• Qualitative variables (nominal, ordinal, and ranked) are attribute or categorical with no intrinsic numerical value.
• The nominal has no order, the ordinal has ordered, and the ranked has observations arrayed in ascending or descending orders of magnitude.

QUANTITATIVE (NUMERICAL) DISCRETE RANDOM VARIABLES - 1:

• The discrete random variables are the Bernoulli, the binomial, the multinomial, the negative binomial, the Poisson, the geometric, the hypergeometric, and the uniform.
• The Bernoulli is the number of successes in a single unrepeated trial with only 2 outcomes.

QUANTITATIVE (NUMERICAL) DISCRETE RANDOM VARIABLES - 1, Con’t...:

• The binomial is the number of successes in more than 2 consecutive trials each with a dichotomous outcome.
• The multinomial is the number of successes in several independent trials with each trial having more than 2 outcomes.

QUANTITATIVE (NUMERICAL) DISCRETE RANDOM VARIABLES - 2:

• The negative binomial is the total number of repeated trials until a given number of successes is achieved.
• The Poisson is the number of events for which no upper limit can be assigned a priori.
• The geometric is the number of trials until the first success is achieved.
• The hypergeometric is the number selected from a sub-sample of a larger sample for example selecting males from a sample of n persons from a population N. The uniform has the same value at repeated trials.

PLOT OF THE BINOMIAL DISTRIBUTION:

PLOT OF THE NEGATIVE BINOMIAL DISTRIBUTION:

PLOT OF THE POISSON DISTRIBUTION:

PLOT OF THE GEOMETRIC DISTRIBUTION: