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

= Conditional probability mass function =
조건부 확률질량함수,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:
 $P_X(x|C)=P(X=x|C)=\frac{P(\{X=x\}\cap C)}{P(C)}$

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.
Sol.
$P(X=2|J_H_1)=\frac{P(\{X=2\}\cap J_H_1)}{P(J_H_1)}=\frac{\frac18}{\frac12}=\frac14.$
$P(X=1|J_H_1)=\frac{\frac38}{\frac12}=\frac34$
$P(X=0|J_H_1)=\frac{0}{\frac12}=0$

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Related:
 [[확률,probability]]
 [[VG:확률,probability]]
 [[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:확률밀도함수,probability_density_function,PDF]]
 [[VG:확률분포,probability_distribution]]
 [[VG:확률질량함수,probability_mass_function,PMF]]
 [[VG:확률함수,probability_function]]
Twin: [[VG:확률및랜덤프로세스]]