Korean, Edit

Chapter 4. Random Variable and Distribution

Higher category : 【Statistics】 Statistics Overview

1. Random variable

2. Distribution function

3. Density function

4. Moment generating function

5. Dimension extension

1. Random variable 

⑴ definition: function or corresponding real value corresponding to each event in the sample space

① X: Ω → ℝ

② F → B(ℝ) = A. B(ℝ): Borel field

③ A ⊂ ℝ, X^-1(A) =｛ω | X(ω) ∈ A｝

④ A ⊂ ℝ, P_X(A) = P(X^-1(A)). P_Xis marked P for convenience

⑵ discrete random variable

① definition 1. when the range of the random variable X is a finite set or a countable infinite set

② definition 2. when the cumulative distribution function F is discrete

⑶ continuous random variable

① definition 1. when the range of the random variable X is an uncountable infinite set 

② definition 2. when the cumulative distribution function F is continuous

⑷ probability distribution : a functional relationship PX that represents the probability of a specific event

① class 1. distribution function: also referred to as cumulative distribution function(CDF)

② class 2. density : categorized by probability mass function and probability density function

③ class 3. moment generating function

④ probability mass function : probability distribution of discrete random variable

⑤ probability density function : probability distribution of continuous random variable

⑥ differentiating the distribution function results in density ↔ integrating the density becomes a distribution function

2. Distribution function

⑴ Definition

① definition: referred to as F(x_i) = P(-∞ ＜ X ≤ x_i) 

② cumulative distribution function of discrete random variable

③ cumulative distribution function of continuous random variable 

⑵ Characteristics 

① F(-∞) = 0, F(∞) = 1

② about x₁ ＜ x ＜ x₂, F(x₁) ≤ F(x₂)

③ regardless of whether the density is a discrete random variable or a continuous random variable, the right limit matches the function value

④ P(a ＜ X ≤ b) = F(b) - F(a) 

⑶ Various probability distributions

3. Density function 

⑴ indication function

① notation : I｛·｝

② definition : a function that is 1 only when · is satisfied and 0 otherwise

⑵ probability mass function (PMF)

① definition: about X =｛x₁, x₂, ···, x_n｝, refers to the function p(x) that satisfies p(x_i) = P(X = x_i)

② example

⑶ probability density function (PDF)

① definition: for the cumulative distribution function F, refers to the function p(x) with F’(x) = p(x)

② example

⑷ common features

① p(x) ≥ 0

② ∫ p(x) dx = 1

⑸ differences

① point probability : probability mass function may not have a point probability of zero, but probability density function always has a point probability of zero

② note that there is a difference between a point probability of 0 and the definition of a support function.

③ support function S_X =｛x | p(x) ＞ 0｝

○ if independent, the support function must be constant for any variable

4. Moment generating function 

⑴ expected value

⑵ moment

① n-th order moment for origin

② 1st order moment for origin : expected value

③ 2nd order moment for origin : moment of inertia

③ 1st order moment for center : 0

④ 2nd order moment for center : variance

⑤ skewness: degree of tilt

○ skew > 0 : When there is a long tail to the right, a lot of data is skewed to the left in response

○ skew < 0 : if there is a long tail to the left

⑥ kurtosis : sharpness

○ similar to normal distribution if the kurtois is close to 3

○ kurtosis ↑: outlayers tend to become crazy values ↑, i.e., tailedness increases

○ kurtosis ↑: more pointed

⑶ moment generating function

① definition

② relationship with the moment

③ similar to Laplace conversion: the momentum generation function and probability distribution are one-to-one correspondence

④ characteristic 1 ψ_aX+b(t) = e^btψ_X(at)

⑤ characteristic 2. If X₁, ···, X_n are independent, about Y = X₁ + ··· + X_n

5. Dimension extension 

⑴ define (X, Y) to have two random variables: Ω → ℝ²

⑵ joint probability distribution (joint probability mass function)

① discrete random variable: about X = ｛x₁, ···, x_m｝, Y = ｛y₁, ···, y_n｝, p(x, y) such that p(x_i, y_j) = P(X = x_i, Y = y_j)

② continuous random variable: a function p(x, y) such that ∂²F(x, y) / ∂x ∂y = p(x, y)

③ characteristic 1. p(x, y) ≥ 0

④ characteristic 2. ∑∑ p(x, y) = 1

⑶ marginal probability distribution

① definition: changing the joint probability distribution to the probability variable X or Y only

② marginal probability distribution of discrete random variables

③ marginal probability distribution of continuous random variables

⑷ example

① knowing the joint probability distribution can tell the distribution of marginal probability

② knowing the marginal probability distribution does not always mean that the joint probability distribution is known

③ Simpson’s paradox: do not calculate intuitively

⑸ joint moment generating function

① definition

② characteristic

⑹ conditional distribution (conditional density function)

① conditional distribution of Y: p (y | x) = p(x, y)／p_X(x)

② conditional distribution of X: p (x | y) = p(x, y)／p_Y(y)

③ application 1.

④ application 2. about Y = g(X), P(X = x) = p(x)

⑤ conditional independence

⑺ mutual independence

① definition

② completeness of the definition of mutually independece 

○ independence can be formed, even in part

○ note that the joint probability distribution and the marginal probability distribution are relative concepts

③ characteristic 1. mutual independence and joint distribution function 

Input : 2019.06.17 13:52