Assignment 9 Solutions

Assignment 9 Solutions - CE93-Engineering Data Analysis...

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Unformatted text preview: CE93-Engineering Data Analysis Prof. Joan Walker, Spring 2011 Assignment 9: Special Distribution and Sampling Statistics Due 4/13/2011 11:10 AM 1. If T has a t-distribution with 8 degrees of freedom, find: a. P(T ≥ 1) Using a t-distribution calculator (or approximating using a t-distribution table…): b. P(T ≤ 2) c. P(-1 ≤ T ≤ 1) 2. Suppose you are waiting for the 51B bus to go to the BART station. The 51B bus arrivals at your bus stop can be approximately be described as a Poisson process with a rate of 1 bus every 12 minutes. a. What is the probability that there are 3 or more 51B bus arrivals in 20 minutes? Let X = # arrivals; X is Poisson random variable with rate = Therefore, = 1.667 b. What is the distribution of the times between 51B bus arrivals? (Hint: It’s a special distribution) Exponential c. What is the variance of the time between 51B bus arrivals? Since time between 51B bus arrivals are exponential… Let T = time between 51B arrivals d. If you just missed the bus, find the probability that you’ll have to wait more than 20 minutes for your next bus. e. If you arrive at the bus stop t minutes after the last bus has departed (t is unknown), find the probability that you’ll have to wait more than 20 minutes for your next bus. f. Based on your results from part a. and b., do Poisson processes “remember” what has happened in the past? This property of the exponential distribution is called memorylessness. Exponential distributions do not “remember” what has happened in the past (nor do Poisson processes in general) CE93-Engineering Data Analysis Prof. Joan Walker, Spring 2011 3. A helicopter is landing at a designated landing area. We are interested in the deviation of the actual helicopter landing position and center of the landing area. If we use the x-y coordinate on the plain with the center of the landing area as the origin, the x-distance between the center of the helicopter landing position and the origin is normally distributed with mean 0 meters and variance 4 m2 and is independent of the y-distance between the center of the helicopter landing position and the origin, which is also normally distributed with mean 0 meters and variance 4 m2. (Hint: Look at example 5.8d for a similarly-structured problem, but use the look-up tables…) a. What is the probability that the distance between the center of the helicopter landing position and the center of the landing area is less than 1.5 m? Let R.V. X denote the x-distance; Y the y-distance X ~ N(0,4); Y ~ N(0,4) Z= – (2 degrees of freedom) b. Suppose the landing area is bounded by a circle. How large the landing area should be such that the helicopter will be able to land fully within the landing area with 99.5% probability? To simplify this problem we assume that the helicopter has a circle shape with a radius of 8 meters. We first look at the distance between the center & the helicopter landing position and the center of the landing area. Suppose this distance must be at least a so as to guarantee that there’s 99% chance that the actual distance is smaller than a. a 8m The subarea delimited by the inner circle denotes the minimum area in which the center of the actual landing position will fall with 99.5% probability. Given the landing position, the minimum landing area should be a circle with radius r + a (as shown by the dashed circles in the figure) Therefore, the landing area should be at least 4. The salary of newly graduated students with bachelor’s degrees in civil engineering has a certain distribution with expected value $53,000, and standard deviation of $3,000. Approximate the CE93-Engineering Data Analysis Prof. Joan Walker, Spring 2011 probability that the average salary of a random sample of 35 recently graduated civil engineers exceeds $54,000. Xi’s are random variables denoting the salary of the ith recently graduated student in the random sample (I = 1, 2, … , 35) 5. The temperature at which a thermostat goes off is normally distributed with variance . If the 2 thermostat is to be tested five times, and S is the sample variance of the five data values, find: a. P( b. P( ) ) c. How large a sample would be necessary to ensure that the probability in part a) is at least .95? We want Try different values for the degrees of freedom in the 2 table within the column = 0.05. We find when , the above inequality is satisfied. Therefore the sample size should be at least 12. 6. A certain component is critical to the operation of a construction machine and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of the components must be in stock so that the probability that the construction machine is in continuous operation for the next 5000 hours is at least 0.99? Let Xi denote the lifetime of the ith component. n components are in stock. CE93-Engineering Data Analysis Prof. Joan Walker, Spring 2011 Therefore: Using quadratic formula, etc… At least 56 components must be in stock. 7. Determine the maximum likelihood estimator of when X 1, … , Xn is a sample with density function: Hint: we order Xi’s from the smallest to the largest, and form a new sequence x1, x2, …, xn. If n is an even number, we pair xi’s as follows: {xi , xn} , {x2 , xn-1} , … , {xk , xk+1}; k = n/2 Note: CE93-Engineering Data Analysis Therefore, when Prof. Joan Walker, Spring 2011 , is achieved 8. Special buoys are used to record the motion of the water surface, in order to obtain measurements of wave heights. From one such buoy we have observed the following wave height measurements (in meters): 3.1 2.0 2.4 3.5 3.2 3.5 4.4 4.5 3.4 2.7 The distribution of ocean wave height (W) can be modeled with the Rayleigh PDF as where is the parameter of the distribution. Find the maximum likelihood estimate for data above. given the ...
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