Читать книгу Informatics and Machine Learning - Stephen Winters-Hilt - Страница 39

Here, we talk of the probability of seeing something after k tries when the probability of seeing that event at each try is “p.” Suppose we see an event for the first time after k tries, that means the first (k − 1) tries were nonevents (with probability (1 − p) for each try), and the final observation then occurs with probability p, giving rise to the classic formula for the geometric distribution:

Figure 2.3 The Geometric distribution, P(X = k) = (1 − p)^(k−1) p, with p = 0.8.

As far as normalization, i.e. do all outcomes sum to one, we have:

Total Probability = ∑_{k = 1}(1 – p)^(k−1) p = p[1 + (1 – p) + (1 – p)² + (1 – p)³ + …] = p[1/(1 − (1 − p))] = 1

So total probability already sums to one with no further normalization needed. In Figure 2.3 is a geometric distribution for the case where p = 0.8: