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k

NN

As N➔∞ get the LLN (weak):

k

N

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:

ssss1 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:

k = 1

So total probability already sums to one with no further normalization needed. In ssss1 is a geometric distribution for the case where p = 0.8:

For the Normal distribution the normalization is easiest to get via complex integration (so we'll skip that). With mean zero and variance equal one (ssss1) we get: