Tutorial 1, Question 1.21

Question

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

Answer 1

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 \(x+2\).

= \(_1p_x\)\(q_{x+1}\)

Comments

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