Properties of PDF (확률밀도함수의 성질)
$\displaystyle \bullet\, f_X(x)\ge 0$ (since CDF is nondecreasing)
$\displaystyle \bullet\, P[a\le X\le b]=\int_a^b f_X(x)dx$
$\displaystyle \bullet\, F_X(x)=\int_{-\infty}^x f_X(t)dt$
$\displaystyle \bullet\, \int_{-\infty}^{+\infty}f_X(t)dt=1$
A valid pdf can be formed by any nonnegative, piecewise continuous function
$\displaystyle g(x)$ that has a finite integral
$\displaystyle \int_{-\infty}^{+\infty} g(x)dx=c<\infty \Rightarrow f_X(x)=g(x)/c$
example: 균등확률변수 uniform r.v.
The pdf of the uniform r.v. is given by
$\displaystyle f_X(x)=\begin{cases}1/(b-a),&a\le x\le b\\0,&{\rm otherwise}\end{cases}$
$\displaystyle \Rightarrow$
$\displaystyle F_X(x)=\begin{cases}0&x<a\\(x-a)/(b-a)&a\le x \le b\\1&x>b\end{cases}$
example:
지수확률함수,exponential_RV
The transmission time X of messages in a communication system has an
지수분포,exponential_distribution:
$\displaystyle P[X>x]=e^{-\lambda x},\;x>0$
이것의 pdf를 구하기
$\displaystyle F_X(x)=P[X\le x]=1-P[X>x]=1-e^{-\lambda x}$
$\displaystyle f_X(x)=\frac{d}{dx}F_X(x)=\frac{d}{dx}(1-e^{-\lambda x})=\lambda e^{-\lambda x}$
(x<0인 경우는 생략)