Chapter 5.1, Problem 7E

A sum of $50000 is invested at a rate R, selected from a uniform distribution on the interval (0.03, 0.07). Once R is selected, the sum is compounded instantaneously for a year, so that $X=50000e^R$ dollars is the amount at the end of that year.

(a) Find the cdf and pdf of X.

(b) Verify that $X=50000e^R$ is defined correctly if the compounding is done instantaneously. Hint. Divide the year into ii equal parts, calculate the value of the amount at the end of each part, and then take the limit as $n\to \infty$.