Discrete case

• Joint pmf of $\underline{X}=(X_1,...,X_n)$

  $P_{\underline{X}} (\underline{x}) = P_{X_1,...,X_n} (x_1,...,x_n) = P(X_1=x_1,...,X_n=x_n)$
  
  properties)
  ① $0 \leq P_{\underline{X}} (\underline{x}) \leq 1$
  ② $\sum_{ \underline{x} \in support} P_{\underline{X}} (\underline{x})=1$

• Joint cdf of $\underline{X}$

  $F_{\underline{X}}(\underline{x}) = F_{X_1,...,X_n} (x_1,...,x_n) = P(X_1 \leq x_1,...,X_n \leq x_n)$

• Marginal pmf of $X_i$

  Let $\underline{X}=(x_1,x_2),$ then
  $P_{X_1} (x_1) = \sum_{x_2} P_{X_1,X_2} (x_1,x_2)$
  $P_{X_2} (x_2) = \sum_{x_1} P_{X_1,X_2} (x_1,x_2)$

• Conditional pmf

  Let $\underline{X}=(x_1,x_2),$ then
  $P_{X_1|X_2} (x_1|x_2) = \frac{P_{X_1,X_2} (x_1,x_2)}{P_{X_2} (x_2)}$

Continuous case

• Joint cdf of $\underline{X}$

  $F_{X_1,...,X_n} (x_1,...,x_n) = P(X_1 \leq x_1,...,X_n \leq x_n)=\int_{-\infty}^{x_1} ... \int_{-\infty}^{x_n} f_{X_1,...,X_n} (t_1,...,t_n) dt_n...dt_1$
  
  Properties) bivariate case
  ① $F(x_1,x_2)$ is non-decreasing in both $x_1$ and $x_2$
  
  ② $\displaystyle \lim_{x_1 \to -\infty} F(x_1,x_2) = \displaystyle \lim_{x_2 \to -\infty} F(x_1,x_2) = 0$
  
  pf) $\displaystyle \lim_{x_1 \to -\infty} F(x_1,x_2) = \int_{-\infty}^{x_2} \int_{-\infty}^{-\infty}f(t_1,t_2) dt_1dt_2 = 0 \space b/c \space \int_{-\infty}^{-\infty}f(t_1,t_2) dt_1 = 0$
  
  ③ $\displaystyle \lim_{x_1 \to \infty} F(x_1,x_2) = F_{X_2} (x_2),\displaystyle \lim_{x_2 \to \infty} F(x_1,x_2) = F_{X_1} (x_1)$
  
  pf) $\displaystyle \lim_{x_1 \to \infty} F(x_1,x_2) = \int_{-\infty}^{x_2} \int_{-\infty}^{\infty}f(t_1,t_2) dt_1dt_2 = \int_{-\infty}^{x_2} f(t_2)dt_2 = F_{X_2}(x_2)$
  
  ④ $F(x_1,x_2)$ is right-continuous in both $x_1$ and $x_2$

• Joint pdf of $\underline{X}$

  If $F$ is differentiable, the joint pdf of $\underline{X}$ is
  $\frac{\partial^n F_{X_1,...,X_n} (x_1,...,x_n)}{\partial x_1...\partial x_n} = f_{X_1,...,X_n} (x_1,...,x_n)$

• Marginal pdf of $X_j$

  $\int_{-\infty}^{\infty} ... \int_{-\infty}^{\infty} f_{X_1,...,X_n} (x_1,...,x_n) dx_1...dx_{j-1}dx_{j+1}...dx_n$

• Conditional pdf

  Let $\underline{X}=(x_1,x_2),$ then
  $f_{X_1|X_2} (x_1|x_2) = \frac{f_{X_1,X_2}(x_1,x_2)}{f_{X_2}(x_2)}$