# for question 9.8 in DHW: how do we know or prove that for independent lives that $$u_{x+t:y+t} = u_{x+t} + u_{y+t}$$

+1 vote
152 views

The question ask to show that the density of $$T_{xy}$$ is $$_tp_{xy}(u_{x+t}+u_{y+t})$$

+1 vote
answered Nov 21, 2017 by (1,390 points)

First, we find the distribution function of $$T_{xy}$$:

$$Pr(T_{xy} \leq t) = 1 - Pr(T_{xy} > t) = 1 - Pr(T_{x} > t) \cdot Pr(T_{y} > t) = 1 - {}_t p_{x} {}_t p_{y}$$.

To find the density, we differentiate the above expression with respect to $$t$$.

$$f_{T_{xy}}(t) = \frac{d}{dt} Pr(T_{xy} \leq t) = {}_t p_{xy}(\mu_{x+t}+ \mu_{y+t})$$ (you need to use the product rule to differentiate the second term in the expression for $$Pr(T_{xy} \leq t)$$).

You will also need expressions of the following form:

$$\frac{d}{dt} {}_t p_{x} = -{}_t p_{x} \mu_{x+t}$$.

Hope this helps.