统计学常见分布计算公式

这是统计学的分布部分的笔记，里面是常见分布汇总

• L 7 - Distribution 1
  • Bernoulli Distribution
    • Definition:
    • n Bernoulli trials
  • Binomial Distribution
    • Definition:
    • deviation
  • Hypergeometric Distribution
    • Definition:
• L 8 - Distribution 2
  • Geometric Distribution
    • Definition:
  • Negative Binomial Distribution
    • Definition:
  • Poisson Distribution
    • Definition:
• L 9 & 10 - Continuous Random Variable 2
  • Uniform Distribution
    • Definition:
  • Exponential Distribution
    • Definition:
  • Normal Distribution
    • Definition:

L 7 - Distribution 1

Bernoulli Distribution

Definition:

A random variable X is said to have a Bernoulli distribution with parameter p, where 0 ≤ p ≤ 1, if its probability mass function is given by

• pmf.: f_X(x) = p^x(1 − p)^{(1 − x)} for x = 0, 1
• mean & expectation: μ = E[X] = p
• variance: σ² = Var[X] = p(1 − p)
• mgf: M_X(t) = E[e^tX] = e^tp + (1 − p), t ∈ (−∞, ∞)

n Bernoulli trials

Definition: a Bernoulli experiment performed n times:

• X₁, X₂, …, X_n are independent Bernoulli random variables (all trials are independent)
• with same parameter p

Binomial Distribution

Definition:

A random variable X is said to have a binomial distribution if its probability mass function is given by

• pmf:
  
• mean & expectation: μ = E[X] = np
  
• variance: σ² = Var[X] = np(1 − p)
  
• mgf: M_X(t) = (pe^t + 1 − p)ⁿ

deviation

if random variables X₁, X₂, …, X_n are independent, then

E[X₁ + X₂ + … + X_n] = E[X₁] + E[X₂] + … + E[X_n]

Var[X₁ + X₂ + … + X_n] = Var[X₁] + Var[X₂] + … + Var[X_n]

M^′(t) = n[(1 − p) + pe^t]^n − 1pe^t ⇒ M^′(0) = E[X] = np

M^″(t) = n(n − 1)[(1 − p) + pe^t]^n − 2p²e^2t + n[(1 − p) + pe^t]^n − 1pe^t

M^″(0) = E[X²] = n(n − 1)p² + np

Var [X] = E[X²] − (E[X])² = n²p² − np² + np − n²p² = np(1 − p)

Hypergeometric Distribution

Definition:

A random variable X is said to have a hypergeometric distribution if its probability mass function is given by

• pmf:
• mean & expectation:
• variance:
• mgf:

L 8 - Distribution 2

Geometric Distribution

Definition:

A random variable X is said to have a geometric distribution if its probability mass function is given by

• pmf: f_X(x) = p(1 − p)^x − 1
• mean & expectation:
• variance:
• mgf:

Negative Binomial Distribution

Definition:

A random variable X is said to have a negative binomial distribution if its probability mass function is given by

• pmf:
• mean & expectation:
• variance:
• mgf:
• negative binomial distribution is a generalization of the geometric distribution

Poisson Distribution

Definition:

A random variable X is said to have a Poisson distribution if its probability mass function is given by

• pmf:
• mean & expectation: μ = E[X] = λ
• variance: σ² = Var[X] = λ
• mgf: M_X(t) = e^{λ(et − 1)}
• Poisson distribution is a limiting case of the binomial distribution when n is large and p is small

L 9 & 10 - Continuous Random Variable 2

第九讲引入了连续随机变量，接着介绍了连续随机变量的分布

Uniform Distribution

Definition:

A random variable X is said to have a uniform distribution if its probability density function is given by

• pdf:
• mean & expectation:
• variance:
• mgf:
• The uniform distribution is often used to model situations where all outcomes are equally likely

Exponential Distribution

Definition:

A random variable X is said to have an exponential distribution if its probability density function is given by

• pdf: f_X(x) = λe^−λx
• mean & expectation:
• variance:
• mgf:

Normal Distribution

Definition:

A random variable X is said to have a normal distribution if its probability density function is given by

• pdf:
• mean & expectation: μ = E[X]
• variance: σ² = Var[X]
• mgf:
• The normal distribution is the most important continuous distribution in probability and statistics