Poisson관련 내용은 [[VG:푸아송_분포,Poisson_distribution]]에 적음.
= 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
조건부 분산,conditional_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:
$P_X(x|C)=P(X=x|C)=\frac{P(\{X=x\}\cap C)}{P(C)}$
(a) the '''conditional expected value''' of X given event C:
$E(X|C)=m_{X|C}=\sum_{\textrm{all }k}x_kP_k(x_k|C)$
(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.
Let r.v. X : the maximum number of heads obtained Tom and Jane each flip a fair coin twice.
||¼||1/16 ||1/8 ||1/16 ||
||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
||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.
$P(X=2|J_H_1)=\frac{P(\{X=2\}\cap J_H_1)}{P(J_H_1)}=\frac{\frac18}{\frac12}=\frac14.$
$P(X=1|J_H_1)=\frac{\frac38}{\frac12}=\frac34$
$P(X=0|J_H_1)=\frac{0}{\frac12}=0$
(c) Find $E(X|J_H_1)$ and $VAR(X|J_H_1).$
||$P(X|J_H_1)$ ||0 ||3/4 ||1/4 ||
$E(X|J_H_1)=0*0+1*(3/4)+2*(1/4)=5/4.$
$VAR(X|J_H_1)=E(X^2|J_H_1)-E(X|J_H_1)^2$
$=(1^2\times\frac34+2^2\times\frac14)-(\frac54)^2=\frac3{16}$
[[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?
$F_X(x)=pF_1(x)+(1-p)F_2(x)$
'''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
$f_X(x)=\begin{cases}1/(b-a),&a\le x\le b\\0,&{\rm otherwise}\end{cases}$
$F_X(x)=\begin{cases}0&x<a\\(x-a)/(b-a)&a\le x \le b\\1&x>b\end{cases}$
example: [[지수확률함수,exponential_RV]]
The transmission time X of messages in a communication system has an [[VG:지수분포,exponential_distribution]]:
$P[X>x]=e^{-\lambda x},\;x>0$
$F_X(x)=P[X\le x]=1-P[X>x]=1-e^{-\lambda x}$
$f_X(x)=\frac{d}{dx}F_X(x)=\frac{d}{dx}(1-e^{-\lambda x})=\lambda e^{-\lambda x}$
||[[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]] ||
[[누적분포함수,cumulative_distribution_function,CDF]]
[[VG:연속확률분포,continuous_probability_distribution]]
[[VG:이산확률분포,discrete_probability_distribution]]
[[VG:전확률정리,total_probability_theorem]]
[[VG:조건부확률,conditional_probability]]
[[VG:집합과_확률,set_and_probability]]
[[VG:확률분포,probability_distribution]]
[[VG:확률함수,probability_function]]
Twin: [[VG:확률및랜덤프로세스]]
Moved to [[확률및랜덤프로세스,probability_and_random_process]]
