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: $\mu =E[X]=p$
• variance: ${\sigma }^2=Var[X]=p(1-p)$
• mgf: $M_X(t)=E[e^{tX}]=e^{t}p+(1-p),t\in (-\infty ,\infty )$

n Bernoulli trials

Definition: a Bernoulli experiment performed n times:

• $X_1,X_2,\dots ,X_n$ 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: $f_X(x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{(n-x)}$
• mean & expectation: $\mu =E[X]=np$
• variance: ${\sigma }^2=Var[X]=np(1-p)$
• mgf: $M_X(t)=(p{e}^t+1-p{)}^n$

deviation

if random variables $X_1,X_2,\dots ,X_n$ are independent, then

$E[X_1+X_2+\cdots +X_n]=E[X_1]+E[X_2]+\cdots +E[X_n]$

$Var[X_1+X_2+\cdots +X_n]=Var[X_1]+Var[X_2]+\cdots +Var[X_n]$

$M^{\prime }(t)=n{\left[(1-p)+p{e}^t\right]}^{n-1}p{e}^t\Rightarrow M^{\prime }(0)=E[X]=np$

$M^{\prime \prime }(t)=n(n-1){\left[(1-p)+p{e}^t\right]}^{n-2}{p}^2{e}^{2t}+n{\left[(1-p)+p{e}^t\right]}^{n-1}p{e}^t$

$M^{\prime \prime }(0)=E[X^2]=n(n-1)p^2+np$

$\mathrm{Var}[X]=E[X^2]-(E[X]{)}^2=n^2{p}^2-n{p}^2+np-n^2{p}^2=np(1-p)$

Hypergeometric Distribution

Definition:

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

• pmf: $f_X(x)=\frac{\left(\genfrac{}{}{0ex}{}{K}{x}\right)\left(\genfrac{}{}{0ex}{}{N-K}{n-x}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)}$
• mean & expectation: $\mu =E[X]=\frac{nK}{N}$
• variance: ${\sigma }^2=Var[X]=n\cdot \frac{K}{N}\cdot \frac{N-K}{N}\cdot \frac{N-n}{N-1}$
• mgf: $M_X(t)={\left(\frac{K}{N}{e}^t+1-\frac{K}{N}\right)}^n$

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: $\mu =E[X]=\frac{1}{p}$
• variance: ${\sigma }^2=Var[X]=\frac{1-p}{p^2}$
• mgf: $M_X(t)=\frac{p{e}^t}{1-(1-p)e^t}$

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: $f_X(x)=\left(\genfrac{}{}{0ex}{}{x-1}{r-1}\right)p^r(1-p{)}^{x-r}$
• mean & expectation: $\mu =E[X]=\frac{r}{p}$
• variance: ${\sigma }^2=Var[X]=\frac{r(1-p)}{p^2}$
• mgf: $M_X(t)={\left(\frac{p}{1-(1-p)e^t}\right)}^r$
• 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: $f_X(x)=\frac{e^{-\lambda }{\lambda }^x}{x!}$
• mean & expectation: $\mu =E[X]=\lambda$
• variance: ${\sigma }^2=Var[X]=\lambda$
• mgf: $M_X(t)=e^{\lambda (e^t-1)}$
• 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: $f_X(x)=\frac{1}{b-a}$
• mean & expectation: $\mu =E[X]=\frac{a+b}{2}$
• variance: ${\sigma }^2=Var[X]=\frac{(b-a{)}^2}{12}$
• mgf: $M_X(t)=\frac{e^{tb}-e^{ta}}{t(b-a)}$
• 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: $f_X(x)=\lambda e^{-\lambda x}$
• mean & expectation: $\mu =E[X]=\frac{1}{\lambda }$
• variance: ${\sigma }^2=Var[X]=\frac{1}{{\lambda }^2}$
• mgf: $M_X(t)=\frac{\lambda }{\lambda -t}$

Normal Distribution

Definition:

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

• pdf: $f_X(x)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$
• mean & expectation: $\mu =E[X]$
• variance: ${\sigma }^2=Var[X]$
• mgf: $M_X(t)=e^{\mu t+\frac{1}{2}{\sigma }^2{t}^2}$
• The normal distribution is the most important continuous distribution in probability and statistics