Categorical Random Variable $x \in {1, 2, \cdots, K}$
Probability Function
Probability Mass Function: returns the probability of a given outcome
Cumulative Distribution Function: returns the probability of a value less than or equal to a given outcome
Percent-Point Function: returns a discrete value that is less than or equal to the given probability
Discrete Probability Distribution
Bernoulli Distribution:
$x \in {0, 1}$
$P(x=1) = p$
$P(x=0) = 1-p$
binom(n=1, p)
Binomial Distribution
Multiple independent Bernoulli trials (Bernoulli process) follows Binomial Distribution
binom(n=100, p=0.3)
Multinoulli Distribution
$x \in {1, 2, \cdots, K }$
$P(x=1) = p_1$
$P(x=2) = p_2$
$P(x=2) = p_2$
$\cdots$
$P(x=K) = p_K$
Multinomial Distribution
Multiple independent Multinoulli trials follows multinomial distribution
multinomial(n=100, p=[0.1, 0.2, 0.7]) #the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on