144 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES
daynwithS 0 given, then
Sn=Sn− 1 +Xn,n≥ 1
whereX 1 ,X 2 ,...are independent, identically distributed continuous
random variables with mean 0 and varianceσ^2. Write an expression
for the probability that you do not recover your purchase price.
- If you buy a lottery ticket in 50 independent lotteries, and in each
lottery your chance of winning a prize is 1/100, write down and evaluate
the probability of winning and also approximate the probability using
the Central Limit Theorem.
(a) exactly one prize,
(b) at least one prize,
(c) at least two prizes.
Explain with a reason whether or not you expect the approximation to
be a good approximation.
- Find a numberksuch that the probability is about 0.6 that the number
of heads obtained in 1000 tossings of a fair coin will be between 440
andk.
- Find the moment generating functionφX(t) =E[exp(tX)] of the ran-
dom variable X which takes values 1 with probability 1/2 and− 1
with probability 1/2. Show directly (that is, without using Taylor
polynomial approximations) thatφX(t/
√
n)n→exp(t^2 /2). (Hint: Use
L’Hospital’s Theorem to evaluate the limit, after taking logarithms of
both sides.)
- A bank has $1,000,000 available to make for car loans. The loans are
in random amounts uniformly distributed from $5,000 to $20,000. How
many loans can the bank make with 99% confidence that it will have
enough money available?
- An insurance company is concerned about health insurance claims.
Through an extensive audit, the company has determined that over-
statements (claims for more health insurance money than is justified by
the medical procedures performed) vary randomly with an exponential
