- The normal (or Gaussian) distribution is denoted by N(μ, σ)
- N(0, 1) is the standard normal distribution
- The z-score of an observation measures how many standard deviation it is away from the mean
- 68-95-99.7 rule: Approximatly 68%, 95% and 99.7% of observations are within 1, 2 and 3 standard deviations of the mean in a standard normal distribution
- Normal probability plots can be used to see how well the normal distribution approximates another distribution
- A random variable with two different outcomes (success or failure, 1 or 0) is called a Bernoulli random variable. Probability for success is denoted by p
- The geometric distribution gives the probability that the k-th successive Bernoulli experiment is the first successful one
- It assumes that the individual experiments / random variables are iid: Independent and identically distributed. This means all variables follow the same probability distribution but don't influence each other
- When performing n iid Bernoulli experiments, the binomial distribution gives the probability that exactly k of them are successful
- Computing the binomial distribution for large values of n becomes very expensive because binomial coefficients grow quickly. The normal distribution gives a good approximation in this case
- However, this approximation breaks down if we are only interested in very small intervals of the distribution
- When performing n iid Bernoulli experiments, the negative binomial distribution gives the probability of the k-th success happening in the very last experiment
- For k = 1 this is the same as the geometric distribution. Otherwise it's the same as multiplying p with the binomial distribution with parameters k - 1, n - 1
- The Poisson distribution helps estimating the number of events in discrete time steps. It's an approximation of the binomial distribution for very large n with a very small p value