By Rainer Winkelmann; Stefan Boes

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P (Ω) = 1. 3. If A1 , A2 , . . , AJ is a ﬁnite or inﬁnite sequence of disjoint events, then ⎞ ⎛ P⎝ J j=1 Aj ⎠ = J P (Aj ) j=1 In practice, we always work with random variables. A random variable Y is a function whose domain of deﬁnition is the set of elementary events ωi and whose range is the set of real numbers. In the above example, we can let Y = 1 if ω =“the person attends university”. A probability model consists then of a sample space, a probability measure, and a random variable. , a set of numerical 30 2 From Regression to Probability Models outcomes together with their probabilities.

3. Multinomial Responses If Y is multinomial distributed with J unordered mutually exclusive outcomes, we could select the probability function of the multinomial distribution. This probability function has J parameters: the outcome probabilities πj for each outcome. The probability function can be written compactly as f (y; π1 , . . , πJ ) = π1d1 π2d2 · · · πJdJ where dj = 1 if y = j and dj = 0 otherwise. Note that the axioms of probability require j pj = 1. Thus, one parameter is deﬁned by the others and we can write J π1 = 1 − πj j=2 In a conditional probability model, the π’s are speciﬁed as functions of X in compliance with the adding-up restriction.

2. Binary Responses Suppose that Y is a binary variable. Obviously, the underlying distribution function is a Bernoulli distribution, which is fully determined by the success probability π. In this case, the counter-probability is 1−π, and the probability function can be written compactly as f (y; π) = π y (1 − π)1−y y = 0, 1 In a conditional probability model, π is speciﬁed as a function of X. 21) in order to make sure that 0 ≤ π(x) ≤ 1 for arbitrary values of x ∈ IR, and without any restrictions on the parameter space of β0 and β1 .