5.8 Key definitions and Relationships

Let $(X,Y)$ be a bivariate rv.

1. 1.

   The joint cdf is $F_{X,Y}(x,y)=𝖯(X\le x,Y\le y)$. $F_X(x)=F_{X,Y}(x,\mathrm{\infty })$.

2. 2.

   For a discrete rv, the joint pmf is $p_{X,Y}(x,y)=𝖯(X=x,Y=y)$.

3. 3.

   For a continuous rv, the joint pdf is $f_{X,Y}(x,y)=\frac{{\partial }^2}{\partial x\partial y}{F}_{X,Y}(x,y)$.

4. 4.

   For discrete rvs $X$ and $Y$, the marginal pmf of $X$, is $p_X(x)={\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}{p}_{X,Y}(x,j)$, and the conditional pmf of $X$ given $Y=y$ is $p_{X|Y}(x|y)=p_{X,Y}(x,y)/p_Y(y)$.

5. 5.

   For continuous rvs $X$ and $Y$, the marginal pdf of $X$ is $f_X(x)={\int }_{t=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{X,Y}(x,t)dt$, and the conditional pdf of $X$ given $Y=y$ is $f_{X|Y}(x|y)=f_{X,Y}(x,y)/f_Y(y)$.

6. 6.

   $X$ and $Y$ are independent if and only if the events $\{X\in A\}$ and $\{Y\in B\}$ are independent for all sets $A$ and $B$: $𝖯(X\in A,Y\in B)=𝖯\left(X\in A\right)𝖯\left(Y\in B\right)$ for all $A$, $B$.

7. 7.

   An equivalent, but easier to check, condition for independence (of discrete or continuous rvs) is: $F_{X,Y}(x,y)=F_X(x)F_Y(y)$. For discrete rvs, independence is also equivalent to $p_{X,Y}(x,y)=p_X(x)p_Y(y)$, whereas for continuous rvs it is equivalent to $f_{X,Y}(x,y)=f_X(x)f_Y(y)$. When just checking factorisation within the range where the rvs are non-zero, variational independence must also be verified.

8. 8.

   Lack of independence can be shown using the two-point method; showing that $f_{X,Y}(x_1,y_1)f_{X,Y}(x_2,y_2)\ne f_{X,Y}(x_1,y_2)f_{X,Y}(x_2,y_1)$ for some $x_1,x_2,y_1,y_2$. Alternatively, show that $f_{X|Y}(x|y)\ne f_X(x)$ for some $y$.