8/18/2019 Problem Set 1stats 2

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Statistics for Financial Engineers, Winter 2015

Professor Martin Lettau

Problem Set 1

Due January 20, 2016 6:00pm

To be submitted via bCourses

1. Let A1, A2,... be an arbitrary infinite sequence of events, and let  B1, B2,...

be another infinite sequence of events defined as follows:   B1  =  A1,  B2  =

Ac1 ∩ A2,  B3  =  A

c1 ∩ Ac

2 ∩ A3,  B4  =  A

c1 ∩ Ac

2 ∩ Ac

3 ∩ A4..... Prove that

P r

  n

i=1Ai

=n

i=1P r(Bi)   for n = 1, 2,...and that

P r

∞i=1

Ai

=

∞i=1

P r(Bi)

2. For any three events   A,   B, and   D, such that   P r(D)   >   0, prove that

P r(A ∪ B|D) =  P r(A|D) + P r(B|D) − P r(A ∩ B|D).

3. Suppose that when a machine is adjusted properly, 50 percent of the items

produced by it are of high quality and the other 50 percent are of medium

quality. Suppose, however, that the machine is improperly adjusted during

10 percent of the time and that, under these conditions, 25 percent of the

items produced by it are of high quality and 75 percent are of medium

quality.

(a) Suppose that five items produced by the machine at a certain time are

selected at random and inspected. If four of these items are of high

quality and one item is of medium quality, what is the probability that

the machine was adjusted properly at that time?

(b) Suppose that one additional item, which was produced by the machine

at the same time as the other five items, is selected and found to be

of medium quality. What is the new posterior probability that the

machine was adjusted properly?

4. Consider a machine that produces items in sequence. Under normal oper-

ating conditions, the items are independent with probability 0.01 of being

defective. However, it is possible for the machine to develop a "memory"

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8/18/2019 Problem Set 1stats 2

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in the following sense: After each defective item, and independent of any-

thing that happened earlier, the probability that the next item is defective

is 2/5. After each nondefective item, and independent of anything that

happened earlier, the probability that the next item is defective is 1/165.

Assume that the machine is either operating normally for the whole time

we observe or has a memory for the whole time that we observe. Let B

be the event that the machine is operating normally, and assume that

P r(B) = 2/3. Let Di  be the event that the ith item inspected is defective.

Assume that D1   is independent of  B .

(a) Prove that P r(Di) = 0.01  for all  i. Hint: use induction.

(b) Assume that we observe the first six items and the event that occurs

is  E  =  Dc

1 ∩ Dc

2 ∩ D3 ∩ D4 ∩ Dc

5 ∩ Dc

6. That is, the third and fourthitems are defective, but the other four are not. Compute  P r(B|E ).

5. Suppose that the PDF of a random variable X   is as follows:

f (x) =

ce

−2x for x > 0,

0   otherwise.

(a) Find the value of the constant c  and sketch the PDF.

(b) Find the value of  P r(1 < X