Information of "event A has high probability value", "event B has medium probability value", "event C has low probability value", ... is called "probability distribution" ================================================== Elementary event (or atomic event): Elementary event has one number of sample $$$P({\text{Diamond}})=0.1$$$ $$$P({\text{Heart}})=0.2$$$ $$$P({\text{Spade}})=0.3$$$ $$$P({\text{Clover}})=0.4$$$ Then, you can calculate probability value of all kinds of events, which has 2 samples, 3 samples, etc according to 3rd rule of Kolmogorov's axiom $$$P({\text{Heart,Spade}})=0.2+0.3=0.5$$$ ================================================== Probability mass function: Probability mass function defines probability values to each elementary event, when there are only finite number of events ================================================== $$$P(\{1\})=0.2$$$ $$$P()$$$ is probability function $$$\{1\}$$$ is event which has one sample $$$0.2$$$ is probability value for event $$$\{1\}$$$ $$$p(1)=0.2$$$ $$$p()$$$ is probability mass function $$$1$$$ is elementary event which has number 1 $$$0.2$$$ is probability value for elementary event 1 ================================================== $$$P(\{1,2\})=0.2$$$ $$$P()$$$ is probability function $$$\{1,2\}$$$ is event which has 2 samples $$$0.2$$$ is probability value for event $$$\{1,2\}$$$ $$$p(1,2)$$$ can be defined ================================================== Simple interval event A: $$$A=\{a\le x < b\}$$$ $$$A = \{ a \leq x < b \} \;\; \rightarrow \;\; P(A) = P(\{ a \leq x < b \}) = P(a, b)$$$ $$$P(B) = P(\{ -2 \leq x < 1\}) + P(\{2 \leq x < 3\}) = P(-2, 1) + P(2, 3)$$$ $$$P(B) = P(\{ -2 \leq x < 3 \}) - P(\{ 1 \leq x < 2\}) = P(-2, 3) - P(1, 2)$$$ ================================================== Cumulative distribution function: From above, you used 2 numbers to define "interval" or "simple interval event A" To use only one number, you can use negative infinity $$$ S_{-1} = \{ -\infty \leq X < -1 \} \\ S_{0} = \{ -\infty \leq X < 0 \} \\ S_{1} = \{ -\infty \leq X < 1 \} \\ S_{2} = \{ -\infty \leq X < 2 \} \\ \vdots \\ S_{x} = \{ -\infty \leq X < x \}$$$ ================================================== In interval $$$\{ a \leq x < b \}$$$ simple intervale probability $$$P(a, b) = P(-\infty, b) - P(-\infty, a)$$$ ================================================== Probability density function: derivative of Cumulative distribution function ================================================== Probability distribution function: Probability mass function Cumulative distribution function Probability density function ==================================================