PMF(Probability Mass Function)

Random variable X: a function that assigns a real number X() to each outcome  in the sample space of a random experiment

sample space S의 사건을 함수 X를 통해 확률변수로 만들어준다

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Classification of Random Variables

이산(Discrete)

연속(Continuous)

cdf

복합(Mixed Type)

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3.2 Probabiltiy Mass Function (pmf)

확률질량함수

X = 변수

x = realization

The pmf of a discrete RV(Random variable) X is defined as:

pX(x) = P[X = x] = P[{ζ: X(ζ) = x}] for x a real number

 

[EX 3.6] A Betting Game

H x 0 and H x 1 → $0

H x 2 → $1

H x 3 → $8

3.3 Expected Value

Expected Value(Mean) of X

$$m_x=E\left[X\right]=\sum _{x\in S_X}^{ }xp_X\left(x\right)=\sum _k^{ }x_kp_X\left(x_k\right)$$

 

* 조건

$$E\left[\left|{X}\right|\right]=\sum _k^{ }\left|{x_k}\right|p_X\left(x_k\right)<\infty $$

 

Expected Value of function

$$Z=g\left(x\right)$$로 두자

$$E\left[Z\right]=E\left[g\left(X\right)\right]=\sum _k^{ }g\left(x_k\right)p_X\left(x_k\right)$$

$$\sum _k^{ }g\left(x_k\right)p_X\left(x_k\right)=\sum _j^{ }z_j\left\{{\sum _{x_k:g\left(x_k\right)=z_j}^{ }p_X\left(x_k\right)}\right\}=\sum _j^{ }z_jp_Z\left(z_j\right)=E\left[Z\right]$$

3.3.2 Variance (분산)

- 분산

$$\sigma ^2=VAR\left[X\right]=E\left[\left(X-m\right)^2\right]=\sum _{ }^{ }\left(x-m_X\right)^2p_X\left(x\right)=E\left[X^2\right]-m_X^2$$

 

- 표준편차

$$\sigma _X=STD\left[X\right]=VAR\left[X\right]^{\frac{1}{2}}$$

 

- 분산성질

 

3.4 Conditional PMF

- 주어진 사건 C의 X의 조건부 pmf

$$p_X\left(x\mid C\right)=p\left[{X=x\mid C}\right]=\frac{P\left[\left\{X=x\ \right\}\cap C\right]}{P\left[C\right]}$$

 

- 성질

$$p_X\left(x\mid C\right)\ge 0,\ \ \forall x$$$$\sum _{x_k\in S_X}^{ }p_X\left(x_k\mid C\right)=1$$$$P\left[X\ in\ B\mid C\right]=\sum _{x\in B}^{ }p_X\left(x\mid C\right),\ \ where\ B\subset S_X$$

 

uniform random variable를 이용하여 m 시간 이후에 m+j 시간에 메세지가 전송될 확률 구하기

 

Conditional Expected value (of given B)

$$m_{X\mid B}=E\left[X\mid B\right]=\sum _{x\in S_X}^{ }xp_X\left(x\mid B\right)=\sum _k^{ }x_kp_X\left(x_k\mid B\right)$$

Conditional Variance

$$VAR\left[X\mid B\right]=E\left[\left(X-m_{X\mid B}\right)^2\mid B\right]=\sum _{k=1}^{\infty }\left(x_k-m_{X\mid B}\right)^2p_X\left(x_k\mid B\right)=E\left[X^2\mid B\right]-m_{X\mid B}^2$$

Total Probabiltiy Theorem
$$P\left[X\right]=\sum _{i=1}^nP\left[X\mid B_i\right]P\left[B_i\right]\ \ \ \to \ E\left[X\right]=\sum _{i=1}^nE\left[X\mid B_i\right]P\left[B_i\right]$$
$$P\left[X\right]=\sum _{i=1}^nP\left[X\mid B_i\right]P\left[B_i\right]\ \ \ \to \ E\left[g\left(X\right)\right]=\sum _{i=1}^nE\left[g\left(X\right)\mid B_i\right]P\left[B_i\right]$$