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

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by (1.5k points)
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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 $$x+2$$.

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

by (1.1k points)

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