Probability distribution

A probability distribution （probability distribution）： Describe the probability of random variables or a cluster of random variables in each possible state . How the probability distribution is described depends on whether the random variables are discrete or continuous .

Discrete variables and probabilistic mass functions

The probability distribution of discrete variables is Probability mass function （probability mass function,PMF）, Usually in capital letters P Express . The probability mass function maps each state that a random variable can obtain to the probability that a random variable can obtain that state .

Joint probability distribution （joint probability distribution）： Probability mass function can act on many random variables at the same time .P(x=x,y=y) Express x=x and y=y Probability of simultaneous occurrence , Or we could just write it as P(x,y).

function P It's a random variable x The probability mass function of , The following conditions must be met ：

• P The domain must be x A collection of all possible states .

• The probability of an unlikely event is 0, And there is no state with a lower probability . Similarly , It can ensure that the probability of certain events is 1, And there is no higher probability than this .

• normalization （normalized） nature , Prevent getting greater than 1 Probability .

Continuous variables and probability density functions

When the object of study is continuous random variable , use Probability density function （probability density function,PDF） Describe its probability distribution .

function P It's a probability density function , The following conditions must be met ：

• p The domain must be x A collection of all possible states .
• Be careful , There is no demand for p(x)≤1.
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