103 - Stochastic Modelling Sept 200 exam Q10 ii)

103 - Stochastic Modelling Sept 2000 exam Q10 ii)

How has this integral been evaluated? I Can not follow these "tricks" that have been used here. I'm guessing by the looks of it the solution is equating the desired integral with the cumulative density function of a normal random variable with standard deviation 1/sqrt(mu)?

Seems to be a very elaborate piece of algebra because usually that integral would require a transformation to polar coordinates.

Can..Not..Follow

 

The integral hasn't really been evaluated. As you guessed they recognised that the (1- the integral) is some multiple of the standard normal distribution function, and made appropriate substitutions.

This "trick" is quite common (maybe more so in CT6).

As far as I know that integral does not have a closed form solution (I think this is the right term).

The polar coordinate transformation works for integrating over all real numbers, and I don't think (although I'm a bit rusty) that it can be applied for integrals over a smaller range.

Since it can't be evaluated algebraically, the common approach (at least for us statisticians/actuaries) is to relate it to the standard normal distribution function (for which we have a convenient pre-calculated table of values, which were obtained via numerical methods). Generally, whenever you need to integrate exp(-x^2), you usually transform it to the standard normal distribution function.

 

I see.

this trick was not discussed in the core reading nor acted text and has not been examined in recent years. Would this really be expected under the current syllabus?

I've noticed the 103 exams contain some very challenging questions as compared to the CT4 exams (on equivalent topics)

 

I suspect that the old 103 may have been a bit more challenging - some of the material would have been inevitably cut when 103 and 104 were joined, although some was pushed to CT5,6,8

I remember using it a bit for the equivalent of CT6 at university for reinsurance type questions. I don't know whether the current CT6 or other subjects utilise this frequently in exams.

Now that you're familiar with the trick, what I can say is that if you do come across an integral which looks difficult, have a glance in the book to see if it's the integral of a standard pdf, which means you might be able to utilise the cdf of that distribution.

If you can easily do all the CT4 papers and handle the 103/104 papers, you'll be in very good form.