It looks like you're presenting a mathematical expression related to logistic regression or some probability model. The notation \( P_i \) typically denotes the probability that a binary outcome \( Y_i = 1 \) given some features \( X_i \). The equation you provided can be interpreted as follows:
The logistic function is often used for modeling the probability of a binary response based on one or more predictor variables.
The logistic function is defined as:
\[ P_i = \text{Pr}(Y_i = 1 | X_i) = \frac{1}{1 + e^{-Z_i}} \]
where \( Z_i \) is a linear combination of the predictors (the features), often expressed as:
\[ Z_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \ldots + \beta_k X_{ik} \]
Here, \( \beta_0 \) is the intercept, and \( \beta_1, \beta_2, \ldots, \beta_k \) are the coefficients associated with the predictors \( X_{i1}, X_{i2}, \ldots, X_{ik} \).
You also seem to be representing the probability in terms of an alternative form:
\[ P_i = \frac{1}{1 + e^{-Z_i}} = \frac{e^{Z_i}}{1 + e^{Z_i}} \]
This transformation holds because multiplying the numerator and denominator by \( e^{Z_i} \) simplifies the expression.
This logistic function maps any real-valued number \( Z_i \) into the (0, 1) range, which makes it suitable for modeling probabilities.
If you have any specific questions or need further explanations about logistic regression or any related topics, feel free to ask!