diff --git a/Presentations/pydata_2023/revenue_retention_presentation.html b/Presentations/pydata_2023/revenue_retention_presentation.html index 4f1a193e..65f5ef19 100644 --- a/Presentations/pydata_2023/revenue_retention_presentation.html +++ b/Presentations/pydata_2023/revenue_retention_presentation.html @@ -8,7 +8,7 @@ - + Cohort Revenue & Retention Analysis: A Bayesian Approach @@ -459,8 +459,8 @@

Example: Shifted Beta Geomet

  1. An individual remains a customer of the company with constant retention probability \(1 - \theta\). This is equivalent to assuming that the duration of the customer’s relationship with the company, denoted by the random variable \(T\), is characterized by the (shifted) geometric distribution with probability mass function and survivor function given by

    -

    \[f(t) = \theta (1 - \theta)^{t - 1}, \quad t = 1, 2, \ldots\]

    -

    \[S(t) = \sum_{j = t}^{\infty} f(j) = (1 - \theta)^{t - 1}, \quad t = 1, 2, \ldots\]

    2. +

    \[f(T=t|\theta) = \theta (1 - \theta)^{t - 1}, \quad t = 1, 2, \ldots\]

    +

    \[S(t) = \sum_{j = t}^{\infty} f(T=j|\theta) = (1 - \theta)^{t}, \quad t = 1, 2, \ldots\]

  2. Heterogeneity in \(\theta\) follows a beta distribution \(\theta \sim \text{Beta}(a, b)\).

diff --git a/Presentations/pydata_2023/revenue_retention_presentation.qmd b/Presentations/pydata_2023/revenue_retention_presentation.qmd index e27ebe57..158f9ec7 100644 --- a/Presentations/pydata_2023/revenue_retention_presentation.qmd +++ b/Presentations/pydata_2023/revenue_retention_presentation.qmd @@ -92,9 +92,9 @@ This is equivalent to assuming that the duration of the customer’s relationshi variable $T$, is characterized by the (shifted) geometric distribution with probability mass function and survivor function given by - $$f(t) = \theta (1 - \theta)^{t - 1}, \quad t = 1, 2, \ldots$$ + $$f(T=t|\theta) = \theta (1 - \theta)^{t - 1}, \quad t = 1, 2, \ldots$$ - $$S(t) = \sum_{j = t}^{\infty} f(j) = (1 - \theta)^{t - 1}, \quad t = 1, 2, \ldots$$ + $$S(t) = \sum_{j = t}^{\infty} f(T=j|\theta) = (1 - \theta)^{t}, \quad t = 1, 2, \ldots$$ 1. Heterogeneity in $\theta$ follows a beta distribution $\theta \sim \text{Beta}(a, b)$.