概率分布：一本直观指南

In this series, I’ll break down the intuition behind the most common probability distributions in Data Science - their structural similarities, use cases and examples.All the distributions we’ll cover are parametric. This just means the entire “shape” of the distribution can be described by a few specific numbers: the parameters. For example, a Normal distribution is defined entirely by just two: its mean and its variance.The underlying event for both of these distributions is a Bernoulli trial. A Bernoulli trial is an event which has one of two outcomes - one is usually labelled success which has the probability, p, and is the event we are interested in, while the other is a failure. Note that the outcome space here is discrete. For example, head in a coin toss, user signing up and etc. A Bernoulli trial on its own isn’t that interesting. It becomes interesting when we do a series of independent Bernoulli trials each with probability of success p. Because then we can ask two questions:What is the probability of getting k successes in n trials? This is modeled by Binomial distribution.What is the probability of getting the first success in k trials? This is modeled by Geometric distribution.Let’s say n = 5 and k = 2. And the probability of success in each independent Bernoulli trial is p.For us to get k = 2 successes, we precisely need 2 successes and 3 failures in 5 trials,\(p ^ 2 (1-p) ^ 3.\)However, note that the 2 successes can come from any 2 of the 5 trials. So for example, in the case of Success, Failure, Failure, Failure, Success, we have, \(P(1-P)(1-P)(1-P)P = p ^ 2 (1-p) ^ 3.\)All of these combinations are captured by the n choose k operator which returns the number of ways we can pick n (=2) successes out of n (=5). So the final probability is, \(\binom{5}{2} p ^ 2 (1-p) ^ 3\)And the generalized form is,\(\binom{n}{k} p ^ k (1-p) ^ {(n-k)}\)We will come back to the binomial distribution later in the series. But as a spoiler, for large values of n the (n choose k) is tedious to calculate, and we will get around that by using the Normal distribution. This approximation is used for running statistical tests on binomial metrics in hypothesis testing.Again, assume that the probability of success in each independent Bernoulli trial is p.So if k = 1, then P is the probability of success in the first trial. Then it is just,\(P(k=1) = p\)For k = 2, the first trial must be a failure and the second one must be a success. And we have,\(P(k=2) = (1-p)p\)For k = 3, \(P(k=3) = (1-p)(1-p)p\)And so on.So this can be generalized for any k, \(P(k) = (1-p)^{(k-1)} p\)Therefore, to compute the probability P of k successes in n independent Bernoulli trials, we use,\(P(k) = \binom{n}{k} p ^ k (1-p) ^ {(n-k)}.\)And to get the probability P of getting the first success in k independent Bernoulli trials, we use,\(P(k) = (1 - p)^{(k-1)}p\)In the next series, we’ll move from discrete counts to time-based continuous models, introducing Poisson processes and the Exponential distribution.

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