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 1. (15 points) Fischer and Spassky play a chess match in which the first player to win a game wins

the match. After 10 successive draws (平局), the match is declared drawn. Each game is won
by Fischer with probability 0.4, is won by Spassky with probability 0.3, and is a draw with
probability 0.3, independent of previous games.
(1) What is the probability that Fischer wins the match? (5 points)
(2) What is the PMF of the duration of the match? (10 points)
2. (15 points) You just rented a large house and the realtor gave you 5 keys, one for each of the 5
doors of the house. Unfortunately, all keys look identical, so to open the front door, you try
them at random. Find the PMF of the number of trials you will need to open the door, under
the following alternative assumptions:
(1) After an unsuccessful trial, you mark the corresponding key, so that you never try it again.
(10 points)
(2) At each trial you are equally likely to choose any key. (5 points)
3. (10 points) Suppose there are 10 electric elements of the same type, among which 2 are
unqualified. You randomly select one from these 10 elements. If it is unqualified, you throw it
away and select one again from the remaining 9 elements; if the second try is still unqualified,
you select another one from the remaining 8 elements. Let 𝑋 be the number of unqualified
elements you select before you get the qualified one. What is the variance of 𝑋?
4. (15 points) Suppose the PDF of a continuous random variable 𝑋 is given by
𝑓(𝑥) = ‘
𝑎𝑥 + 𝑏𝑥!, 0 < 𝑥 < 1,
0, otherwise.
If the expectation of 𝑋 is E(𝑋) = 2/3, please calculate Var(𝑋).
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5. (10 points) Let 𝑋 be a Poisson random variable with parameter 𝜆. Show that the PMF 𝑝”(𝑘)
increases monotonically with 𝑘 up to the point where 𝑘 reaches the largest integer not
exceeding 𝜆, and after that point decreases monotonically with 𝑘.
6. (10 points) Continued with Example 3.8 in the slides of Chapter 2 – Part 1, calculate the
probability that the insurance company loses money in this life insurance, i.e., the profit is less
than $0.
7. (10 points) Jobs are sent to a printer independently at a constant rate of 3 jobs per hour.
(1) What is the expected time between jobs? (5 points)
(2) Suppose that a job just arrived, what is the probability that the next job is sent within 5
minutes? (5 points)
8. (15 points) Continued with Page 41 of Chapter 2 – Part 2, please write down the relationship
between the geometric distribution and the exponential distribution with detailed derivation.