Modelo Geométrico

Understanding Geometric Distributions

Introduction to Geometric Models

• The discussion begins with an introduction to discrete models, specifically focusing on the geometric model. It highlights that the variable x is a real number and p is a parameter within the range of 0 to 1, similar to binomial distributions.

Characteristics of Geometric Distribution

• Two cases are presented regarding the geometric model, emphasizing how the range of the random variable varies while still relating back to the parameter p . The concept of success and failure in this distribution is reiterated.

• The geometric distribution is defined as counting failures until achieving the first success. This definition sets up how mass functions are structured around these probabilities.

Mass Function Definition

• The mass function for a geometric distribution indicates that there’s one probability for success needed to conclude an experiment, multiplied by probabilities representing failures leading up to that success.

Expectation and Variance

• A theorem states that if X follows a geometric distribution with parameter p , then its expectation (mean) is given by q/p , where q = 1 - p , and variance is given by q/p^2 .

Different Forms of Geometric Distribution

• There’s mention of another form where if X sim G(1,p) , it counts trials until achieving the first success rather than counting failures. This distinction clarifies why counting starts from 1 instead of 0 in this case.

• The necessity for at least one trial before reaching a success in this version contrasts with earlier definitions where zero trials could lead directly to success without any failures recorded.

Probability Representation Changes

• As trials increase (e.g., needing two trials for one success), changes in representation occur within the mass function based on whether it’s defined as starting from zero or one.

Relationships Between Versions

• Finally, relationships between both versions of geometric distributions are discussed, indicating how they can be interrelated through transformations such as defining new variables based on existing ones.