46

Probability, Random Variables and Expectations

Exercises
Exercise 1.1. Prove that E [a + b X ] = a + b E [X ] when X is a continuous random variable.
Exercise 1.2. Prove that V [a + b X ] = b 2 V [X ] when X is a continuous random variable.
Exercise 1.3. Prove that Cov [a + b X , c + d Y ] = b d Cov [X , Y ] when X and Y are a continuous random variables.
Exercise 1.4. Prove that V [a + b X + c Y ] = b 2 V [X ] + c 2 V [Y ] + 2b c Cov [X , Y ] when X and Y are a continuous random variables.
¯
Exercise 1.5. Suppose {X i } is an sequence of random variables. Show that V X = V
2
2 σ where σ is V [X 1 ].

time.

1.4 Expectations and Moments

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i. Assuming 99% of trades are legitimate, what is the probability that a detected trade is rogue?
Explain the intuition behind this result. ii. Is this a useful test? Why or why not?
Exercise 1.13. You corporate finance professor uses a few jokes to add levity to his lectures. He is also very busy, and so forgets week to week which jokes were used.
i. Assuming he has 12 jokes, what is the probability of 1 repeat across 2 consecutive weeks? ii. What is the probability of hearing 2 of the same jokes in consecutive weeks? iii. What is the probability that all 3 jokes are the same? iv. Assuming the term is 8 weeks long, and they your professor has 96 jokes, what is the probability that there is no repetition across the term? Note: he remembers the jokes he gives in a particular lecture, only forgets across lectures.
v. How many jokes would your professor need to know to have a 99% chance of not repeating any in the term?
Exercise 1.14. A hedge fund company manages three distinct funds. In any given month, the probability that the return is positive is shown in the following table:
Pr
Pr
Pr
Pr

r1,t r2,t r3,t r2,t >
>
>
>

0 = .55
0 = .60
0 = .45
0 ∩ r3,t >

Pr r1,t
Pr