Revision Questions

These questions relate to the maths that (I think) you already know. If not, don’t worry - have another look at the Prerequisites document.

Exercise 0.1 Suppose \(X \sim \cN(\mu, \sigma^2)\) and \(Y = a + bX\) (\(b \neq 0\)). Find \(\E(Y)\), \(\Var(Y)\) and \(\Cov(X, Y)\). What is the distribution of \(Y\)?

Exercise 0.2 In each round of a game you win \(0.15\) with chance \(0.55\) or lose \(0.1\) with chance \(0.45\). Successive rounds are independent and you play 1,000 times. Use the Central Limit Theorem to estimate the chance that your total winnings exceed 35.

Exercise 0.3 Suppose that over a unit period the price of a stock changes from \(s\) to either \(us\) with chance \(p\) or \(ds\) with chance \(1-p\). Successive price changes are independent. Estimate the probability that the stock price will increase by at least 30% over the next 100 time periods when \(u = 1.02\), \(d = 0.98\) and \(p = 0.52\).

Hint: switch to \(\log\) scale and compare with the previous problem.

Exercise 0.4 Show that, if \(a \geq 0\) is constant, \((1 + a/n)^n \to e^a\) as \(n \to \infty\).

Hint: analyse the behaviour of \((1 + a/x)^x\) as \(x \to \infty\) by taking \(\log\) and applying Taylor’s theorem (treating the remainder with professional care!). Alternatively, using the definition of \(\log\), show that if \(n > -a\) we have \(a/(n+a) < \log(1+a/n) < a/n\) and deduce the result from this.

Exercise 0.5 You take out a loan for the amount \(L\). For the next \(n\) months you will pay back amount \(a\) at the end of each month. Interest accrues at rate \(r\) per month – that is, if you owe \(R_j\) at the end of month \(j\), you will owe \((1+r)R_j\) at the end of the next month (before your next payment is made).

Express the amount \(a\) that leads to the loan being paid back exactly in terms of \(L\), \(n\) and \(r\).

Hint: let \(R_j\) be the amount owed at the end of month \(j\) (after the payment, so \(R_0 = L\) and \(R_n = 0\)) and relate \(R_j\) to \(R_{j+1}\).

Check that \(a > rL\). Why is this essential? How much of the payment at the end of month \(j\) was for interest and how much for paying off the loan principal? Check that the amounts paid off the principal do add up to \(L\).