# How to compute the joint probability function of two discrete random variables given the joint distribution table

I am trying to find the value $$P( X > 6, Y < 7)$$. The main difficulty is that I cannot apply the cdf formula
\begin{align}F_{xy}( x_1 < X < x_2 \cap y_1 < Y < y_2) &= F_{xy}(x_2,y_2) - F_{xy}(x_1,y_2) - F_{xy}(x_2,y_1) \\&+ F_{xy}(x_1,y_1) \end{align} as I would have $$F_{xy}(-\infty,+\infty)$$. Another possible way would be to sum the components in the joint distribution table where the properties are met.

• Could you explain where the infinities come in? They suggest that somehow there are infinite values of $X$ and $Y$ listed in your table, which seems unlikely. – whuber Feb 2 at 18:51
• I assumed that if $x_1 < X < x_2$ and $x_1 = 6$ then $x_2 = +\infty$; – Sergiu Talmacel Feb 2 at 19:30

You can write what you ask (instead of $$<$$, you'll need $$\leq$$ whenever CDF is concerned, especially with discrete RVs) in terms of CDFs: $$P(X>6,Y\leq7)=P(Y\leq 7)-P(X\leq 6, Y\leq 7)=F_Y(7)-F_{XY}(6,7)$$
And, your formula is correct only when $$X\leq x_2$$ and $$Y\leq y_2$$. And, you can calculate the marginal CDF of $$Y$$, using the joint CDF and the supremum $$X$$ value that can occur for $$x$$.
• First to second is from total probability i.e. $$P(X\leq 6\cap Y\leq 7)+P(X>6\cap Y\leq 7)=P(Y\leq 7)$$ second to third is just definiton of CDF. – gunes Feb 2 at 19:35