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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. Probability generating function

6. Dimension extension

1. Random variable 

⑴ Definition: A function that maps each event in the sample space to a real number, or the real value mapped by the function.

① A random variable is usually denoted by a capital letter X, and each value is denoted by x or xi.

② Example 1. Die: {1, ···, 6} → ℝ with Die(i) = i

③ Example 2. Coin: {head, tail} → ℝ with Coin(head) = 1, Coin(tail) = 0

④ Example 3. Sum: {(i, j)

i, j = 1, ···, 6} with Sum(i, j) = i + j

⑤ X : Ω → ℝ

⑥ F → B(ℝ) = A, B(ℝ): Borel field

⑦ A ⊂ ℝ, X-1(A) = ｛ω

X(ω) ∈ A｝

⑧ A ⊂ ℝ, P_X(A) = P(X^-1(A)); P_X is denoted as P for convenience.

⑨ Example problems of the concept of random variables

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

Figure 1. meaning of skewness

○ 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

Figure 2. Meaning of Kurtosis

○ 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

⑧ Example problems for moment

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

⑥ Example problems for moment generating function

5. Probability generating function

⑴ Definition: It is primarily defined for discrete random variables.

⑵ Relationship with moment generating function

⑶ Sum of independent random variables and probability generating functions

⑷ n-th factorial moment: Denoted as μ_(n), it can be expressed as the n-th derivative of the probability generating function.

⑸ Probability generation: p_n = Pr(N = n) can be understood as the coefficient of xⁿ in the Taylor series expansion.

⑹ Example problems for probability generating functions

6. 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

⑤ Example problems for joint probability distribution

⑥ Example problems for advanced joint probability distribution

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

④ Example problems for marginal probability distribution

⑤ Example problems for advanced marginal probability distribution

⑷ Example

Table 1. examples of joint probability distribution and marginal probability distribution

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

Table 2. Simpson’s paradox

⑸ Joint moment

① Example problems for joint moment

② Example problems for advanced joint moment

⑹ Joint moment generating function

① Definition

② Characteristic

⑹ Conditional distribution

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 probability expression for independence

⑥ Conditional independence

⑥ Example problems of conditional distribution

⑧ Example problems for joint conditional distribution

⑨ Example problems for joint conditional moment

⑻ 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