Difference between r1.8 and the current
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#noindex
Sub:[[확률변수,RV]]
[[무기억성,memoryless_property]]
1. 확률 ¶
Related:
확률,probability - 확률,probability
확률론,probability_theory =확률론,probability_theory =,probability_theory 확률론 probability_theory
{
확률론
확률론
probability_theory
}
확률,probability - 확률,probability
확률론,probability_theory =확률론,probability_theory =,probability_theory 확률론 probability_theory
{
확률론
확률론
probability_theory
}
연속확률분포,continuous_probability_distribution
연속확률분포,continuous_probability_distribution
연속확률분포
continuous_probability_distribution
연속확률분포,continuous_probability_distribution
연속확률분포
continuous_probability_distribution
이산확률분포,discrete_probability_distribution
이산확률분포,discrete_probability_distribution
이산확률분포
discrete_probability_distribution
이산확률분포,discrete_probability_distribution
이산확률분포
discrete_probability_distribution
(이상 둘은 비교대상임, mk table)
확률분포,probability_distribution
확률분포,probability_distribution
probability_distribution
Probability_distribution ?
확률분포
확률분포
확률분포,probability_distribution
probability_distribution
Probability_distribution ?
확률분포
확률분포
확률함수,probability_function
probability_function ?
probability_function ?
확률함수,probability_function
확률함수
probability_function ?
probability_function ?
확률함수,probability_function
확률함수
3. Conditional probability mass function ¶
조건부 확률질량함수,conditional_pmf
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
Conditional + 확률질량함수,probability_mass_function,PMF
조건부확률질량함수,conditional_probability_mass_function,conditional_PMF
조건부 기대치,conditional_expected_value조건부확률질량함수,conditional_probability_mass_function,conditional_PMF
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 PX and event C with P(C)>0.
→ the conditional probability mass function of X given event C:
(a) the conditional expected value of X given event C:
pmf 자리에 조건부pmf가 왔음.
→ the conditional probability mass function of X given event C:
$\displaystyle P_X(x|C)=P(X=x|C)=\frac{P(\{X=x\}\cap C)}{P(C)}$
def.(a) the conditional expected value of X given event C:
$\displaystyle E(X|C)=m_{X|C}=\sum_{\textrm{all }k}x_kP_k(x_k|C)$
사건 C가 일어났을 때 X의 조건부 기대치.pmf 자리에 조건부pmf가 왔음.
(b) the conditional variance of X given event C:
$\displaystyle VAR(X|C)=E\left((X-m_{X|C})^2\right)=E(X^2|C)-\left(E(X|C)\right)^2$
C라는 사건이 일어났을 때 X의 조건부 분산.4. 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.
이것은
ⓐ 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:
9. Cumulative Distribution Function , Probability Density Function
Sol.
J\T | 0 | 1 | 2 |
0 | 0 | 1 | 2 |
1 | 1 | 1 | 2 |
2 | 2 | 2 | 2 |
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 한번 나오는 것은, 앞뒤 뒤앞이니까 1/2
ⓒ 앞앞이니까 1/4
so, pmf:
X | 0 | 1 | 2 |
PX | 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 $\displaystyle J_{H_1}$ 로 표기.)
Sol.
$\displaystyle P(X=2|J_{H_1})=\frac{P(\{X=2\}\cap J_{H_1})}{P(J_{H_1})}=\frac{\frac18}{\frac12}=\frac14.$
$\displaystyle P(X=1|J_{H_1})=\frac{\frac38}{\frac12}=\frac34$
$\displaystyle P(X=0|J_{H_1})=\frac{0}{\frac12}=0$
(사건 "Jane got one head in two tosses"를 $\displaystyle J_{H_1}$ 로 표기.)
Sol.
$\displaystyle P(X=2|J_{H_1})=\frac{P(\{X=2\}\cap J_{H_1})}{P(J_{H_1})}=\frac{\frac18}{\frac12}=\frac14.$
$\displaystyle P(X=1|J_{H_1})=\frac{\frac38}{\frac12}=\frac34$
$\displaystyle P(X=0|J_{H_1})=\frac{0}{\frac12}=0$
(c) Find $\displaystyle E(X|J_{H_1})$ and $\displaystyle VAR(X|J_{H_1}).$
Sol. Since
$\displaystyle E(X|J_{H_1})=0*0+1*(3/4)+2*(1/4)=5/4.$
$\displaystyle VAR(X|J_{H_1})=E(X^2|J_{H_1})-E(X|J_{H_1})^2$
$\displaystyle =(1^2\times\frac34+2^2\times\frac14)-(\frac54)^2=\frac3{16}$
X | 0 | 1 | 2 |
$\displaystyle P(X|J_{H_1})$ | 0 | 3/4 | 1/4 |
$\displaystyle VAR(X|J_{H_1})=E(X^2|J_{H_1})-E(X|J_{H_1})^2$
$\displaystyle =(1^2\times\frac34+2^2\times\frac14)-(\frac54)^2=\frac3{16}$
이상 이산, 이후 연속
5. 누적분포함수,cdf ¶
누적분포함수,cumulative_distribution_function,CDF
def. For r.v. X, the CDF of X:
right-continuous, staircase function with jumps
$\displaystyle F_X(x)=\sum_{x_k\le x}p_X(x_k)=\sum_k p_X(x_k) u(x-x_k)$
$\displaystyle F_X(x)=P(X\le x),\quad\quad -\infty
이산확률변수,discrete_RV의 CDFright-continuous, staircase function with jumps
$\displaystyle F_X(x)=\sum_{x_k\le x}p_X(x_k)=\sum_k p_X(x_k) u(x-x_k)$
연속확률변수,continuous_RV의 CDF
continuous, nonnegative function $\displaystyle f(x)$ 의 적분으로 쓸 수 있음
$\displaystyle F_X(x)=\int_{-\infty}^x f(t)dt$
continuous, nonnegative function $\displaystyle f(x)$ 의 적분으로 쓸 수 있음
$\displaystyle F_X(x)=\int_{-\infty}^x f(t)dt$
Mixed R.V의 CDF..... 이게 뭐람?
$\displaystyle F_X(x)=pF_1(x)+(1-p)F_2(x)$
$\displaystyle F_X(x)=pF_1(x)+(1-p)F_2(x)$
6. 확률밀도함수,pdf ¶
PDF는 CDF의 미분,derivative으로 정의.
For 연속확률변수,continuous_RV:
$\displaystyle f_X(x)=\frac{dF_X(x)}{dx}$
For 이산확률변수,discrete_RV:$\displaystyle f_X(x)=\frac{d}{dx}\sum_k p_X(x_k)u(x-x_k)=\sum_k p_X(x_k)\delta(x-x_k)$
참고로 delta function $\displaystyle \delta(t):$$\displaystyle u(x)=\int_{t=-\infty}^x \delta(t)dt$
see 디랙_델타함수,Dirac_delta_functionProperties of PDF (확률밀도함수의 성질)
The pdf of the uniform r.v. is given by
The transmission time X of messages in a communication system has an 지수분포,exponential_distribution:
$\displaystyle \bullet\, f_X(x)\ge 0$ (since CDF is nondecreasing)
$\displaystyle \bullet\, P[a\le X\le b]=\int_a^b f_X(x)dx$
$\displaystyle \bullet\, F_X(x)=\int_{-\infty}^x f_X(t)dt$
$\displaystyle \bullet\, \int_{-\infty}^{+\infty}f_X(t)dt=1$
A valid pdf can be formed by any nonnegative, piecewise continuous function $\displaystyle g(x)$ that has a finite integral$\displaystyle \bullet\, P[a\le X\le b]=\int_a^b f_X(x)dx$
$\displaystyle \bullet\, F_X(x)=\int_{-\infty}^x f_X(t)dt$
$\displaystyle \bullet\, \int_{-\infty}^{+\infty}f_X(t)dt=1$
$\displaystyle \int_{-\infty}^{+\infty} g(x)dx=c<\infty \Rightarrow f_X(x)=g(x)/c$
example: 균등확률변수 uniform r.v.The pdf of the uniform r.v. is given by
$\displaystyle f_X(x)=\begin{cases}1/(b-a),&a\le x\le b\\0,&{\rm otherwise}\end{cases}$
$\displaystyle \Rightarrow$$\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_RVThe 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)$\displaystyle f_X(x)=\frac{d}{dx}F_X(x)=\frac{d}{dx}(1-e^{-\lambda x})=\lambda e^{-\lambda x}$
8. Textbooks ¶
Probability, Random Variables and Random Signal Principles, 4th Edition
Peyton Peebles Jr
Peyton Peebles Jr
Probability and Random Processes with Application to Signal Processing
Stark and Woods
Stark and Woods
이상은 학부과정, 이하는 대학원과정
An Introduction to Statistical Signal Processing
Gray and Davisson
Gray and Davisson
Probability, Random Variables, and Stochasitc Processes, 4th edition
Papoulis and Pillai
Papoulis and Pillai