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

---

Classification of Random Variables

이산(Discrete)

연속(Continuous)

cdf

복합(Mixed Type)

---

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

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

 

* 조건

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

 

Expected Value of function

$$Z=g\left(x\right)$$로 두자

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

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

3.3.2 Variance (분산)

- 분산

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

 

- 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

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

 

- 성질

$$p_X(x\mid C)\ge 0,\mathrm{\forall }x$$$$\sum _{x_k\in S_X}^{}{p}_X(x_k\mid C)=1$$$$P[XinB\mid C]=\sum _{x\in B}^{}{p}_X(x\mid C),whereB\subset S_X$$

 

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

 

Conditional Expected value (of given B)

$$m_{X\mid B}=E[X\mid B]=\sum _{x\in S_X}^{}x{p}_X(x\mid B)=\sum _k^{}{x}_k{p}_X(x_k\mid B)$$

Conditional Variance

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

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