STAT 2655 Assignment 3 - Due in class November 18, 2015 1) One model for plant competition assumes that there is a zone of resource depletion around each plant seedling. Depending on the size of the zones and the density of the plants, the zones of resource depletion may overlap with those of other seedlings in the vicinity. When the seeds are randomly dispersed over a wide area, the number of neighbours that any seedling has within an area of size A usually follows a Poisson distribution with mean equal to A d, where d is the density of seedlings per unit area. Suppose that the density of seedlings is four per square meter. What is the probability that a specied seeding has: (a) No neighbours within 1 meter (b) At most three neighbours within 2 meters? (c) At least 3 neighbours within area A. Suppose you know that A, the area of interest, is the region bounded between f1 (x) = x2 2x and f2 (x) = 4 x2 . 2) The number of cars driving past a parking area in a one-minute time interval has a Poisson distribution with mean . The probability that any individual driver actually wants to park their car is p. Assume that individual drivers decide whether they want to park independently of one another. (a) If one parking space is a available and it will take you one minute to reach the parking lot, what is the probability that the space will still be available when you reach the lot? (Assume nobody leaves the lot during the one minute interval.) (b) Let W denote the number of drivers who wish to park during a one-minute interval. Derive the probability distribution of W . 3) A gas station operates two pumps, each of which can pump up to 10,000 liters of gas in a month. The total amount of gas pumped at the station in a month is a r.v. X with pdf (measured in 10,000 liters) given by: , 0x<1 x cx , 1x<2 f (x) otherwise. (a) find the value of c. (b) graph (x). (c) cdf and it. (d) probability that station will pump between 8000 12000 liters in a particular month. (e) given pumped more than 10000 month, nd it 15000 liters. (f) show e[x] =10000. stat 2655 4) suppose y is continuous random variable with density (y), 0. if (y) cdf, e[y ] =0 hint:> 0 then y = y 0 [1 F (y)]dy. yf (y)dy = 0 1dt and E[Y ] = 0 y 0 1dt f (y)dy 5) Weekly CPU time used by an accounting rm has a pdf (measured in days) given by f (x) = 3 2 x (4 64 0 x) , 0 x < 4 , otherwise. (a) Find the expected value and variance of weekly CPU time. (b) CPU time costs the rm $200 per day. Find the expected cost and variance of the cost of weekly CPU time. (c) Would you expect the weekly cost to exceed $600 very often? Why? 6) Beginning at midnight, a server is up for one hour and down for two hours on a regular cycle for cost savings. A person who is unaware of this schedule tries to access the server at a random time between midnight and 5:00 am. What is the probability that the server is running when a person's request comes in? 7) The length of life of oil drilling bits depends on the type of rock and soil that the drill encounters, but it is estimated that the mean length of life is 75 hours. An oil exploration company purchases drill bits whose length of life is approximately normally distributed with mean 75 hours and standard deviation 12 hours. What proportion of the companies drill bits: (a) will fail before 60 hours of use? (b) will last at least 60 hours? (c) will have to be replaced after more than 90 hours of use? 8) Let X denote the r.v. with pdf given by 1 f (x) = e|x| , < x < . 2 Find its mgf m(t) and use it to nd E[X] and V [X]