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2023 FIN2210                Problem Set 11

Probability for Finance               

1. Suppose  is an IID random sample from a normal population  Find the limiting distribution of the sample mean  as .

2. One observation in taken on a discrete random variable X with PMF , where . Find the MLE of θ

3. Let X1, ..., Xn be i.i.d. with one of the following two pdfs. If  = 0, then

  while if  = 1, then

Find the MLE of .

4. Suppose that the random variables satisfy

where  are fixed constants, and  is an i.i.d. sequence from a

 ,  are unknown.

(a) Find the MLE of , and show that it is an unbiased estimator of

(b) Find the distribution of MLE of

5. Let be an IID Bernoulli(p) random sample and define .

(1) Show that .

(2) Show that for , the estimator  of the population variance satisfies.

6. Suppose  is an IID  random sample. What is the asymptotic distribution of  

7. One observation, X, is taken from a  population.

(a) Find an unbiased estimation of :

(b) Find the MLE of .

(c) Discuss how the method of moments estimator of  might be found.

8. Suppose {X1, X2, ..., Xn} is an i.i.d. random sample from some population with unknown mean  and variance: Define parameter .

(a) Suppose  is an estimator for , where  is the sample mean. Show that  is not unbiased for θ. (Hint: )

(b) Find an unbiased estimator for θ.