## Friday, 21 February 2014

### Lecture 16

The human life plots of force of mortality that I showed in today's lecture can be viewed on the Understanding Uncertainly page: How long are you going to live?

David Spiegelhalter (Winton Professor for the Public Understanding of Risk) has now added a comment for us in the comments section at the end of this post. He makes reference to the Gompertz distribution, which is one with a linearly increasing hazard rate.

Notice that the hazard rate curve (force of mortality) has a 'bath-tub' shape - there is an early high-risk period immediately following birth, followed by long low period of annual risk, until accidents start kicking in for 17-year-old-boys, and then steadily increases. At the above link there are some tools for interactively exploring how your life expectancy changes as a function of your present age, gender and lifestyle (non-smoker, moderate alcohol, 5 a day, exercise).

Most things (such as car engines, tyres, light bulbs) have an increasing hazard rate. But some things have a decreasing hazard rate. A business that has lasted two centuries is less likely to go bankrupt next year than one that only started two years ago.

Notice that if $X\sim U[0,1]$ then $h(x)=f(x)/(1-F(x))=1/(1-x)$, which tends to $\infty$ as $x\to 1$.

Some fine fine print. I said that a continuous random variable is one that has a continuous c.d.f., and it is this that puts the "continuous" in its name. However, the fine print (which you can find in my notes near the middle of page 63) is that we actually need the c.d.f. to absolutely continuous, and this will ensure that a p.d.f. exists. The qualifier "absolutely" rules out weirdly pathological c.d.fs. For example, the Cantor distribution has a continuous c.d.f., but it has no p.d.f. Of course none of this fine print is examinable in Part IA. But I know that some of you find this type of thing interesting.