Sub:
 [[확률변수,RV]]
 [[무기억성,memoryless_property]]

Poisson관련 내용은 [[VG:푸아송_분포,Poisson_distribution]]에 적음.

[[TableOfContents]]

= 확률 =
Related:
[[확률,probability]] - [[VG:확률,probability]]
[[확률론,probability_theory]] =확률론,probability_theory =,probability_theory 확률론 probability_theory
{
확률론
Ndict:확률론 
WtEn:probability_theory
}

[[연속확률분포,continuous_probability_distribution]]
[[VG:연속확률분포,continuous_probability_distribution]]
Ndict:연속확률분포
WtEn:continuous_probability_distribution

[[이산확률분포,discrete_probability_distribution]]
[[VG:이산확률분포,discrete_probability_distribution]]
Ndict:이산확률분포
WtEn:discrete_probability_distribution

(이상 둘은 비교대상임, mk table)

[[전확률정리,total_probability_theorem]]
[[VG:전확률정리,total_probability_theorem]]
Ndict:전확률정리
Bing:전확률정리

[[조건부확률,conditional_probability]]
[[VG:조건부확률,conditional_probability]]
Ndict:조건부확률
WtEn:conditional_probability

[[VG:집합과_확률,set_and_probability]]
.... Ggl:집합+확률+비교+표 Naver:집합+확률+비교+표 Bing:집합+확률+비교+표

[[확률분포,probability_distribution]]
[[VG:확률분포,probability_distribution]]
WtEn:probability_distribution
WpSp:Probability_distribution ?
Ndict:확률분포
확률분포

[[확률함수,probability_function]]
WtEn:probability_function ?
WpSp:probability_function ?
[[VG:확률함수,probability_function]]
Ndict:확률함수

= 랜덤프로세스 =
random process = stochastic process


= Conditional probability mass function =
조건부 확률질량함수,conditional_pmf
 Conditional + [[VG:확률질량함수,probability_mass_function,PMF]]
 조건부확률질량함수,conditional_probability_mass_function,conditional_PMF
조건부 기대치,conditional_expected_value
 [[VG:기대값,expected_value]]
조건부 분산,conditional_variance
 [[VG:분산,variance]]

from http://www.kocw.net/home/search/kemView.do?kemId=1279832 8. Conditional Probability, Independence of Events, Sequential Experiments 1:02:39

Let d.r.v.(discrete random variable) X with pmf P,,X,, and event C with P(C)>0.
→ the '''conditional probability mass function''' of X given event C:
 $P_X(x|C)=P(X=x|C)=\frac{P(\{X=x\}\cap C)}{P(C)}$

def.
(a) the '''conditional expected value''' of X given event C:
 $E(X|C)=m_{X|C}=\sum_{\textrm{all }k}x_kP_k(x_k|C)$
사건 C가 일어났을 때 X의 조건부 기대치.
pmf 자리에 조건부pmf가 왔음.

(b) the '''conditional variance''' of X given event C:
 $VAR(X|C)=E\left((X-m_{X|C})^2\right)=E(X^2|C)-\left(E(X|C)\right)^2$
C라는 사건이 일어났을 때 X의 조건부 분산.

= ex. =
Let r.v. X : the maximum number of heads obtained Tom and Jane each flip a fair coin twice.

(a) Find the pmf of X.
Sol.
||J＼T ||0 ||1 ||2 ||
||0 ||0 ||1 ||2 ||
||1 ||1 ||1 ||2 ||
||2 ||2 ||2 ||2 ||
이것은
||곱 ||¼ⓐ||½ⓑ||¼ⓒ||
||¼||1/16 ||1/8 ||1/16 ||
||½||1/8  ||1/4 ||1/8  ||
||¼||1/16 ||1/8 ||1/16 ||
ⓐ tom이 앞면의 개수가 0이 나오는 것은, 1/2 * 1/2 = 1/4
ⓑ tom이 앞면의 개수가 한번 나오는 것은, 앞뒤 뒤앞이니까 1/2
ⓒ 앞앞이니까 1/4
so, pmf:
||X ||0 ||1 ||2 ||
||P,,X,, ||1/16 ||1/2 ||7/16 ||

9. Cumulative Distribution Function , Probability Density Function

(b) Find the conditional pmf of X=2 given that Jane got one head in two tosses.
(사건 "Jane got one head in two tosses"를 $J_{H_1}$ 로 표기.)
Sol.
$P(X=2|J_{H_1})=\frac{P(\{X=2\}\cap J_{H_1})}{P(J_{H_1})}=\frac{\frac18}{\frac12}=\frac14.$
$P(X=1|J_{H_1})=\frac{\frac38}{\frac12}=\frac34$
$P(X=0|J_{H_1})=\frac{0}{\frac12}=0$

(c) Find $E(X|J_{H_1})$ and $VAR(X|J_{H_1}).$

Sol. Since
||X ||0 ||1 ||2 ||
||$P(X|J_{H_1})$ ||0 ||3/4 ||1/4 ||
$E(X|J_{H_1})=0*0+1*(3/4)+2*(1/4)=5/4.$
$VAR(X|J_{H_1})=E(X^2|J_{H_1})-E(X|J_{H_1})^2$
$=(1^2\times\frac34+2^2\times\frac14)-(\frac54)^2=\frac3{16}$

이상 이산, 이후 연속

= 누적분포함수,cdf =
[[VG:누적분포함수,cumulative_distribution_function,CDF]]
 
def. For r.v. X, the CDF of X:
 $F_X(x)=P(X\le x),\quad\quad -\infty<x<\infty.$

[[이산확률변수,discrete_RV]]의 CDF
right-continuous, staircase function with jumps
$F_X(x)=\sum_{x_k\le x}p_X(x_k)=\sum_k p_X(x_k) u(x-x_k)$

[[연속확률변수,continuous_RV]]의 CDF
continuous, nonnegative function $f(x)$ 의 적분으로 쓸 수 있음
$F_X(x)=\int_{-\infty}^x f(t)dt$

Mixed R.V의 CDF..... 이게 뭐람?
$F_X(x)=pF_1(x)+(1-p)F_2(x)$

= 확률밀도함수,pdf =
'''PDF'''는 CDF의 [[미분,derivative]]으로 정의.

For [[연속확률변수,continuous_RV]]:
 $f_X(x)=\frac{dF_X(x)}{dx}$

For [[이산확률변수,discrete_RV]]:
 $f_X(x)=\frac{d}{dx}\sum_k p_X(x_k)u(x-x_k)=\sum_k p_X(x_k)\delta(x-x_k)$

참고로 delta function $\delta(t):$
 $u(x)=\int_{t=-\infty}^x \delta(t)dt$

see [[VG:디랙_델타함수,Dirac_delta_function]]

Properties of PDF (확률밀도함수의 성질)
 $\bullet\, f_X(x)\ge 0$ (since CDF is nondecreasing)
 $\bullet\, P[a\le X\le b]=\int_a^b f_X(x)dx$
 $\bullet\, F_X(x)=\int_{-\infty}^x f_X(t)dt$
 $\bullet\, \int_{-\infty}^{+\infty}f_X(t)dt=1$

A valid pdf can be formed by any nonnegative, piecewise continuous function $g(x)$ that has a finite integral
 $\int_{-\infty}^{+\infty} g(x)dx=c<\infty \Rightarrow f_X(x)=g(x)/c$

example: 균등확률변수 uniform r.v.
The pdf of the uniform r.v. is given by
 $f_X(x)=\begin{cases}1/(b-a),&a\le x\le b\\0,&{\rm otherwise}\end{cases}$
$\Rightarrow$
 $F_X(x)=\begin{cases}0&x<a\\(x-a)/(b-a)&a\le x \le b\\1&x>b\end{cases}$


example: [[지수확률함수,exponential_RV]]
The transmission time X of messages in a communication system has an [[VG:지수분포,exponential_distribution]]:
 $P[X>x]=e^{-\lambda x},\;x>0$
이것의 pdf를 구하기
 $F_X(x)=P[X\le x]=1-P[X>x]=1-e^{-\lambda x}$
 $f_X(x)=\frac{d}{dx}F_X(x)=\frac{d}{dx}(1-e^{-\lambda x})=\lambda e^{-\lambda x}$
(x<0인 경우는 생략)

= 또 다른 정리 시도... =
||[[이산확률변수,discrete_RV]] ||[[연속확률변수,continuous_RV]] ||
||[[VG:이산확률분포,discrete_probability_distribution]] ||[[VG:연속확률분포,continuous_probability_distribution]] ||
||[[VG:확률질량함수,probability_mass_function,PMF]] ||[[VG:확률밀도함수,probability_density_function,PDF]] ||



[[VG:누적분포함수,cumulative_distribution_function,CDF]]

= Textbooks =
Probability, Random Variables and Random Signal Principles, 4th Edition
Peyton Peebles Jr

Probability and Random Processes with Application to Signal Processing
Stark and Woods

Leon-Garcia - see [[VG:확률및랜덤프로세스]]

이상은 학부과정, 이하는 대학원과정

An Introduction to Statistical Signal Processing
Gray and Davisson

Probability, Random Variables, and Stochasitc Processes, 4th edition
Papoulis and Pillai

via 조준호 https://youtu.be/c0o8P-QveuQ?t=817

----
Twin: [[VG:확률및랜덤프로세스]]