Distribution

random variable: map random event to real number.

given any random event, measure the possibility.

we have many methods to represent the distribution:

• Cumulative Density Function: $F(x)=P(X<x),P([a,b])=F(b)-F(a)$
• Possibility Density Function: $f(x),P([a,b])={\int }_a^{b}f(x)dx$
• Possibility Mass Function: $P(x=k)$

Probability Density Function

$$f(x)=P^{\prime }(X\le x)$$

Cumulative Distribution Function

$$P(X\le x)$$

according to Fundamental Theorem of Calculus, we can briefly define CDF as

$$P(X\le x)={\int }_{-\infty }^{x}f(t)dt$$

Thus:

$${\int }_{-\infty }^{\infty }f(x)dx=P(X\le +\infty )=1$$

Transformation

$Y=X+C$

$$\begin{array}{rl}& P(Y<y)=P(X<y-C)={\int }_{-\infty }^{y-C}f(t)dt={\int }_{-\infty }^{y}f(s-C)ds\\ ⟹& f_Y(y)=P^{\prime }(Y<y)=f(y-C)\end{array}$$

$Y=CX,C>0$

$$\begin{array}{rl}& P(Y<y)=P\left(X<\frac{y}{C}\right)={\int }_{-\infty }^{y/C}f(t)dt={\int }_{-\infty }^{y}f\left(\frac{s}{C}\right)d\left(\frac{s}{C}\right)=\frac{1}{C}{\int }_{-\infty }^{y}f\left(\frac{s}{C}\right)ds\\ ⟹& f_Y(y)=P^{\prime }(Y<y)=\frac{1}{C}f\left(\frac{y}{C}\right)\end{array}$$