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2023

FIN2210

Problem Set 10

Probability for Finance                

1. Suppose  for some constant .  is a constant.

(1) Does  converge to  in  norm? If so, prove it. If not, give an example.

(2) Prove that  using the epsilon-delta argument. Do not simply apply Lemma 5 on the textbook.

2. .

(1) Show that . Be careful when you calculate the  norm, i.e., .

(2) Show that .

3. Suppose a sequence of random variables  is defined as

(1) Does Zn converge in mean square (i.e., ) to 0? Give your reasoning clearly.

(2) Does Zn converge in probability to 0? Give your reasoning clearly.

4. Let the sample space S be the closed interval [0,1] with the uniform probability distribution. Define Z(s) = s for all  Also, for  define a sequence of random variables

(1) Does Zn converge in quadratic mean to Z?

(2) Does Zn converge in probability to Z?

(3) Does Zn converge almost surely to Z?

5. Suppose  are a sequence of independent N(0,1) random variables. Define

Find the limiting distribution of  as  and explain your reasoning.

6. Suppose the death rate of a certain life insured is 0.005. Now we have 1000 people buy this insurance. Use the CLT to estimate:

(1) the probability of 40 people died within a year.

(2) the probability of less than 70 people died within a year.