Chapter 4

보통 Sample space를 로 나타냄. Random variable X=1 이나 Y=3 이런것들은 변수를 말하는게 아니라 함수와 같은 역할을 함.

Remark

는 Probability function.

2. Types of random variable

Cumulative distribution function

Probability mass function

주사위나 동전 던지기 같은 것. Sample space가 유한이 아니라 countable이기만 하면 됨.

Probability mass function 이라고 함.

을 만족.

Probability density function

키, 몸무게, 어떤 실수든지 변수가 될 수 있는것을 Sample space로 가짐. e.g. 그런데, . 즉, 특정 점의 확률은 0이 됨. 마치 화살과녁에 특정 점에 화살을 맞추는건 불가능한 것처럼.

가 continuous random variable이면, 인 실수 전역에서 nonnegative한 가 존재한다. 이 함수가 바로 Probability density function 이다.

e.g. . 가 p.d.f. 임 성질들은 다음과 같음.

Examples

Let X be continuous random variable and p.d.f

Now find

3. Jointly distributed random variables

Joint cummulative probability distribution function

Joint p.m.f.

of X and Y

가 discrete random variables 이면,

similar in

Joint p.d.f

of X and Y

are jointly continuous if there exists for all real and such that note that

Recall joint cumulative probability distribution function .

Notation :

Independent random variables

Definition

Suppose that X and Y are independent. Then,

B. Conditional distributions

Recall that conditional probability .

If X and Y are discrete random variables,

Conditional p.m.f.

of for given

If and are jointly continuous and have joint probability density function ,

Conditional p.d.f.

of for given

4. Expectation

Notation.

If X is discrete random variable, If X is continuous random variable,

유도과정

for .

5. Properties of the expected value

여기서 은 합성함수 가 아니라, 일때 인 함수를 뜻한다.

Proposition 1.

Expectation of a function of a random variable.

• X가 discrete random variable 일때.
• X가 continuous random variable 일때.

Corollary 2.

If constants, then

proof. If discrete, If continuous,

By this,

A. Expected Value of sums of random variables

For example, if ,

In general,

6. Variance

Definition.

is random variable with mean , then the variance of is

For any constants

7. Covariance and variance of sums of random variables

Definition.

Covariance of X, Y. Notation :

성질들

3. for any constant

Lemma 1.

Thm.

일반적으로, 이다. 성립은 가 independent 일때만 가능.

Proposition 2.

Corollary 3.

8. Moment generating function

Definition

Moment generarting function of X := Hence Hence

Let , then Hence,

9. Chebyshev’s inequality and the weak law of large numbers

Proposition 1. (Markov’s inequality)

X is random variable that takes only nonnegative values

Proposition 2. (Chebyshev’s inequality)

If X is a random variable with mean and variance ,

Thm 3. (The weak law of large numbers)

Let be the sequence of independent an identically distributed random variables, each having mean . Then, for any ,