# Tutorial 1, Question 1.21

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Hi.

I am struggling with Q1.21 for the first tutorial. I can do part (a) but I cannot understand how to do (b)

1.21 Suppose that deaths are uniformly distributed between consecutive integer ages. This problem sets out to show, step by step, that Tx−Kx then has exactly a uniform distribution on the interval (0,1).

$$f_x$$ denotes the pdf of $$T_x$$

Show that for $$1\leq t < 2$$ , $$f_x(t)$$= $$_tp_x$$ $$q_{x+1}$$

+1 vote
answered Mar 26, 2017 by (1,490 points)
selected Mar 26, 2017 by Rowan

Hi Rohin

You seemed to have either misread or mistyped the question. We need to prove for $$1\le t \lt 2$$, $$f_x(t)$$ = $$_1p_x$$$$q_{x+1}$$.

From part (a), we proved for $$0 \le t\lt 1$$, $$f_x(t)$$ = $$q_x$$ using the assumption that deaths are uniformly distributed.

For $$1\le t\lt 2$$, we have:

$$f_x(t)$$ = $$S_x(1)$$ $$f_{x+1}(t-1)$$

This is as a result of thinking of the pdf of $$T_x$$ for $$1 \le t \lt 2$$ as the probability of someone surviving from age $$x$$ to age $$x+1$$ multiplied by the probability of someone dying within the period from age $$x+1$$ to $$x+2$$.

= $$_1p_x$$$$q_{x+1}$$

commented Mar 26, 2017 by (1,040 points)

Intuitively it makes a lot of sense. However, is there a way we could prove it? (if necessary)