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A logistic regression model can be expressed in several ways:

Logistic Regression Function:

The logistic regression function is given by:
[
P(Y=1|X) = \frac{1}{1 + e^{-(\beta0 + \beta1X)}}
]
Here, (P(Y=1|X)) is the probability that the binary response (Y) is 1 given the predictor (X), (\beta0) is the intercept, and (\beta1) is the slope parameter.

Logit Function (Log-Odds):

The logit function, which is the natural logarithm of the odds, is expressed as:
[
\text{logit}(P) = \log\left(\frac{P}{1-P}\right) = \beta0 + \beta1X ​
]
This linearizes the logistic regression model, making it easier to interpret the parameters.

Odds:

The odds of the event occurring (i.e., (Y=1)) can be expressed as:
[
\text{Odds} = \frac{P}{1-P} = e^{\beta0 + \beta1X}
]
This shows how the odds change with a unit change in the predictor (X). ​

Probability:

The probability of the event occurring can be directly expressed using the logistic function:
[
P(Y=1|X) = \frac{e^{\beta0 + \beta1X}}{1 + e^{\beta0 + \beta1X}}
]