# Using joint density to calculate probability that one person dies before the other

Let $$T_x, T_y$$ be two random variables that describe the remaining lifetime of two persons aged $$x$$ and $$y$$, respectively. The joint density of $$T_x$$ and $$T_y$$ is given by

$$f_{T_x,T_y}(s,t)=\begin{cases}\frac{2}{45}\cdot20^{-4}\cdot\Big(9 \cdot 20^2-(3s-t)^2\Big)&,s \in [0,20], t\in[0,60]\\0&, \mathrm{otherwise}\end{cases}$$

How can I use this to determine the probability that the person aged $$y$$ dies before the person aged $$x$$?

## 2 Answers

You need to integrate the joint density over the region where $$T_y$$ is less than $$T_x$$. I suggest drawing a picture. Your integration region is the rectangle $$(s,t) \in [0,20]\times[0,60]$$ where $$s$$ corresponds to $$T_x$$ and $$t$$ corresponds to $$T_y$$. If you draw the line $$s = t$$ you should be able to work out the limits of integration from there.

• So what are my limits of integration? – user636610 Jan 26 at 17:30
• I added some additional hints above. Short answer: draw a picture and it should be clear. – inhuretnakht Jan 26 at 18:24

Draw a rectangle boundary for the joint PDF, where x axis in $$[0,20]$$ and y axis is $$[0,60]$$. We need $$P(T_y. $$y$$ is smaller than $$x$$ below $$y=x$$ line. So, you need to integrate the joint in the region bounded by the lines $$y=x,\ \ y=0, \ \ x=20$$ as follows:

$$P(T_y