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The Bernoulli distribution is a fundamental probability distribution in statistics and probability theory. It models a random experiment with two possible outcomes: success (usually denoted as 1) and failure (usually denoted as 0). The distribution is named after Swiss mathematician Jacob Bernoulli (refer to page3887).

The Bernoulli distribution is characterized by a single parameter, which is the probability of success, often denoted as "p." The probability mass function (PMF) of the Bernoulli distribution is given by:

          ------------------- [3865a]

Where:

• P(X = x) is the probability that the random variable X takes the value x (either 0 or 1).
• p is the probability of success or event (the parameter of the distribution). The parameter "p" represents the probability of success in a single trial of the experiment. It is the parameter that governs the shape and behavior of the Bernoulli distribution.
• x is the observed outcome (either 0 or 1).

Based on Equation 3865a, we can have,

          ------------------- [3865b]

          ------------------- [3865c]

Then, comparing with the Exponential Family equation below:

          ----------------------- [3865d]

we have:

          h(x) = 1

          θ = log(p/(1-p))

          x = T(x)

          A(θ) = log(1-p) = -log(1-1/(1+exp(-θ)))

The probability distribution of the dependent variable in logistic regression follows:

          1) A Bernoulli distribution for binary logistic regression.

          2) A categorical distribution for multinomial logistic regression.

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