Let X and Y be two random variables defined on the same sample space S. Then the function (X,Y) that assigns a point in \mathbb{R*R} is called a two dimensional random variable.
Examples:
- Taking measurement of height and weight of people
- Recording ages of husband and wife of couples.
Joint CDF of (X,Y): The joint cdf of (X,Y) is defined as:
F_{X,Y}(x,y) = Pr \left[ X \leq x , Y \leq y \right] \quad ,\forall (x,y) \in \mathbb{R^2}
Properties of CDF:
\begin{matrix} 1. & F_{X,Y}(-\infty,y)= lim_{x \rightarrow -\infty} F_{X,Y}(x,y) =0 , \forall y \\ & F_{X,Y}(x,-\infty)= lim_{y \rightarrow -\infty} F_{X,Y}(x,y) =0 , \forall x \\ 2. & F_{X,Y}(\infty,\infty)= lim_{x,y \rightarrow \infty} F_{X,Y}(x,y) =1 & \\ 3. & \hskip{-4.5cm} \text{For any } h>0 , k>0, & \\ & Pr \left[ x < X < x+h , y < Y < y+k \right] &\\ &= F(x+h,y) - F(x,y+k) - F(x+h,y) +F(x,y) \\ 4. & F(x,y) \text{ is continuous from the right in each argument, that is} &\\ & lim_{\epsilon \rightarrow 0^+} F(x+\epsilon,y) = lim_{\epsilon \rightarrow 0^+} F(x,y+\epsilon) =F(x,y) \end{matrix}
Proof of Property 1:
Suppose, A_{n,y} = \{ -\infty < X \leq -n , -\infty < Y \leq y \} \\ ;\text{ where } n \text{ is a positive integer. Clearly, } A_{n,y} \downarrow as \quad n \uparrow. \\ Thus, \quad lim_{n \rightarrow \infty} A_{n,y} = \phi \\ Now, \\ \begin{matrix} F(-\infty,y) =lim_{n \rightarrow \infty} F(-n,y) &=& lim_{n \rightarrow \infty} P(A_{n,y}) \\ &=& P(lim_{n \rightarrow \infty} A_{n,y} ) \\ & & \text{[by continuity theorem]} \\ &=& P (\phi) =0 \end{matrix} \\ \text{Similary, the other half of the property can be proved.}
PROOF OF PROPERTY 2:
Suppose, B_n=\{ -\infty < X \leq n , -\infty < Y \leq n \} . Clearly, B_n \uparrow as \\ n\uparrow. Thus, lim B_n = \Omega .\text{Hence the proof.}
PROOF OF PROPERTY 3:
\begin{matrix} && Pr \left[ x < X \leq x+h , y < Y \leq y+k \right] \\ &=& F(x+h,y+k) - F(x+h,y) - F(x,y+k) + F(x,y) \\ &\geq &0 \end{matrix}
PROOF OF PROPERTY 4:
Suppose, C_n=\{ -\infty < X \leq x+ \frac{1}{n} , -\infty < Y \leq y \} \\ \text{,where n is a positive integer.Then, } \\ \underset{n \rightarrow \infty }{lim} F(x+1/n,y) = F(x,y)
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