p, we can expect first
success in 1/p trials P(A) * P(B/A)
P(A/B) = ──────────────
P(B)
—
Derivationʳ:
P(A ∩ B) is the probability of A times probablity of B
given that A has already happened.
P(A ∩ B) = P(A) * P(B/A)
It could also be defined as the probability of B times probablity of A given that B has already happened.
P(A ∩ B) = P(B) * P(A/B)
Equating the two,
P(A) * P(B/A) = P(B) * P(A/B)
P(A) * P(B/A)
=> P(A/B) = ──────────────
P(B)
Same event that happens multiple times over a time interval.
Probability of k events (probability density/mass function):
λᵏ.e⁻ᵏ
P(k) = ───────
k!
Poisson density function is not continuous. It's defined only for
integer values of k.
See: https://brilliant.org/wiki/poisson-distribution/
DBT: Mean is past the midpoint in the graph always??
Value of random variable is from one among a set of predefined categories ʳ
Each category has an associated probability
A generalization of Bernoulli distribution ?? ʳ
Discrete
Doesn't have anything to do with category theory
Examples:
Given a collection of points (of any probability distribution. Need not be normal), if we select k number of points repeatedly with replacement (ie, the k points are considered to be 'put back' after each trial), the mean value of the trials will be normally distributed.
See: