AMS 311: Introduction to Probability Sample Final Exam
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AMS 311: Introduction to Probability
Sample Final Exam
1. A number is selected at random from the set of natural numbers f1, 2, 3, . . . , 1000000g. What is the probability that it is not divisible by 5, 7, or 8?
Answer: 0.6.
2. In a town, 7/9 of the men and 3/5 of the women are married. In that town, what fraction of the adults are married? Assume that all married adults are the residents of the town.
Answer: 21/31.
3. A box contains 20 fuses, of which ive are defective. What is the expected number of defective items among three fuses selected randomly?
Answer: 0.75.
4. Suppose that X is a discrete random variable with E[X] = 1 and E[X(X - 2)] = 3. Find Var(-3X + 5).
Answer: 36.
5. Suppose that X is a continuous random variable satisies
P (X > t) = αe- t + βe-μt , t ≥ 0,
where α + β = 1, α ≥ 0, β ≥ 0, λ > 0, μ > 0. Compute E[X].
Answer: α + .
6. X and Y have joint density
f (x, y) = C(y - x)e-g , - y < x < y, 0 < y < 1.
Find E[X].
Answer: -1.
7. A isherman catches ish in a large lake with lots of ish. The number of ish caught during time t has a Poisson distribution with expectation 2t. In particular, this implies that on average he catches 2 ish per hour. The time he spends ishing on a given day is uniformly distributed between 3 and 8 hours. Find the variance of the number of ish he catches.
Answer: 11 + .
8. On each bet, a gambler loses 1 with probability 0.7, loses 2 with probability 0.2, or wins 10 with proba- bility 0.1. Approximate the probability that the gambler will be loosing after his irst 100 bets.
Answer: 0.6104.
2023-12-15