ax, 0 < x < 1 0, otherwise. f(x)dx = 1).

Transcription

1 HW 3 (Due Oct. 3, 2017) Name: HW 3.1 Suppose that X has the pdf f(x) = { ax, 0 < x < 1 0, otherwise. (a) Find the value of a (use the requirement f(x)dx = 1). (b) Calculate P (X < 0.3) (c) Calculate P (0.3 X < 0.8) (d) Find the mean and variance of X (e) If we define Y = 3X, calculate the mean and variance of Y. 1

2 HW 3.2. Astronomers treat the number of stars in a given volume of space as a Poisson random variable. The density in the Milky Way Galaxy in the vicinity of our solar system is one star per 16 cubic light-years. (a) What is the probability of two or more stars in 16 cubic light-years? (b) How many cubic light-years of space must be studied so that the probability of one or more stars exceeds 0.95? 2

3 HW 3.3. A particularly long traffic light on your morning commute is green on 20% of the mornings. Assume that each morning represents an independent trial. (a) What is the probability that the first morning that the light is green is the fourth morning? (b) What is the probability that the light is not green for 10 consecutive mornings? (c) What is the probability that no more than two green lights will be observed among the first 3 mornings? 3

4 HW 3.4. A utility company might offer electrical rates based on time-of-day consumption to decrease the peak demand in a day. Enough customers need to accept the plan for it to be successful. Suppose that among 50 major customers, 15 would accept the plan. The utility selects 10 major customers randomly (without replacement) to contact and promote the plan. (a) What is the probability that exactly two of the selected major customers accept the plan? (b) What is the probability that at least one of the selected major customers accepts the plan? 4

5 HW 3.5. Printed circuit cards are placed in a functional test after being populated with semiconductor chips. A lot contains 140 cards, and 20 are selected without replacement for functional testing. (You should clearly define your random variable, identify the type of its distribution with correctly parameters, and then write down what probability the question is asking, finally, calculate the probability) (a) If 20 cards are defective, what is the probability that at least 1 defective card is in the sample? (b) If 5 cards are defective, what is the probability that at least 1 defective card appears in the sample? 5

6 HW 3.6. An article in Ad Hoc Networks [ Underwater Acoustic Sensor Networks: Target Size Detection and Performance Analysis (2009, Vol.7(4), pp )] discussed an underwater acoustic sensor network to monitor a given area in an ocean. The network does not use cables and does not interfere with shipping activities. The arrival of clusters of signals generated by the same pulse is taken as a Poisson arrival process with rate of λ per unit time. Suppose that for a specific underwater acoustic sensor network, this Poisson process has a rate of 2.5 arrivals per unit time. (a) What is the expected time (i.e., mean) between 2.0 consecutive arrivals? What is the variance? (b) What is the probability that the time until the first arrival exceeds 1.0 unit of time? (c) Denote the time from the second arrival to the third arrival to be X, find P (X > 10 X > 9). 6

7 HW 3.7. If X is normally distributed with a mean of 5 and a standard deviation of 4. (a) Find P (2 < X < 8), P (X > 10), P (X < 9). (b) Determine the value for x that solves each of the following: (1) P (X > x) = 0.5 (2) P (X > x) = 0.95 (3) P (x < X < 9) = 0.2 (4) P (3 < X < x) = 0.95 (5) P ( x < X 5 < x) =

8 HW 3.8. (a) Suppose s arrive according to a Poisson process with rate being 10 s per hour. Let X denote the time between the 11th and 21st arrival. What is the distribution of X. (b) Suppose s arrive according to a Poisson process with rate being 10 s per hour. Let X denote the time until the first arrival, find P (X > 2 X > 1) (c) Suppose X denotes the salary of new graduated students, and it follows a normal distribution with mean 50000, and variance 20000, find the value of x such that P (X > x) =

9 HW 3.9. X = time (in seconds) to react to brake lights during in-traffic driving. We assume X N(µ = 2, σ 2 = 0.25). (a) suppose that we take a random sample of n = 5 drivers with times X 1, X 2,..., X 5. What is the distribution of the sample mean X? Is X an unbiased estimator of µ? Find P (X > 2.1). (b) Suppose that we take a random sample of n = 25 drivers with times X 1, X 2,..., X 25. What is the distribution of the sample mean X? Find P (X > 2.1). 9

10 HW The time to death for rats injected with a toxic substance, denoted by X (measured in days), follows an exponential distribution with λ = 1/4. That is, X Exp(λ = 1/4). Suppose that we take a random sample of n = 36 drivers with times X 1, X 2,..., X 5. approximate) P (X > 3). Find (or 10