Texts: 2) [This one is a little involved, but quite important and illustrates the usefulness of large sample theory to guide decisions. See the video example provided on the course webpage for a very similar problem].
Consider question 1 above and the given monthly profit density function. Suppose you plan to stay in business for 30 years. You wish to approximate the probability that your combined profit over this period will exceed $400,000. First, set up the problem, like the video describes, and write down mathematically the probability that you seek. Then, manipulate this probability into a form where the central limit theorem can be applied to provide an approximation of the probability that total profits will exceed $400,000. (Note: Be careful and recall that profits are measured in $1,000s of dollars. Also, use the result in (1a) and (1b) of question 1).
Note: You will need to make use of the normal cdf here. To do so, you can either use the table provided on the course website or use Matlab. (Or both if you wish to check your answer). Within Matlab, normcdf() returns the value of the normal cdf at the point. For example, normcdf(0) will return 0.5 as the answer.
Just for fun (at least I think it's fun!), I have written a separate MATLAB program to tackle this question in a different way. Specifically, I generated 360 draws from the monthly profit distribution (take more econometrics and you will learn how!) and calculated the total profit over this period. I then repeated this process 100,000 times, obtaining a total 30-year profit value each time, thus tracing out a simulated distribution of total profits over a 30-year horizon. A plot of that distribution is provided below. Note that 77.8 percent of the total profit simulations I obtained were larger than $400.
Profits (in $1,000s) over 30 years:
