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homework_05 - are statistically independent what is the...

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MIT OpenCourseWare http://ocw.mit.edu 1.010 Uncertainty in Engineering Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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% % 1.010 Fall 2008 Homework Set #5 Due October 16, 2008 (in class) 1. Two continuous variables X and Y have joint probability density function: # 1 f X , Y ( x , y ) = $ 8 ( x + y ) 0 " x " 2 and 0 " y " 2 & 0 elsewhere a) Find and plot the marginal probability density function (PDF) of X . b) Find and plot the marginal cumulative distribution function (CDF) of X . c) Find and plot the conditional PDF of ( Y | X =1). d) Are X and Y independent? Comment. 2. According to schedule, Train A arrives at station S at 10:55 am and Train B departs from the same station at 11:05 am. Due to delays, the arrival time of Train A is uniformly distributed between 10:55 and 11:10 and the departure time of Train B is uniformly distributed between 11:05 and 11:15. If the arrival and departure times of the two trains
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Unformatted text preview: are statistically independent, what is the probability that a passenger on Train A misses the connection with Train B?[Hint: Let T A and T B be the times when train A arrives and train B departs, respectively. Plot the joint range of ( T A , T B ) on the ( T A , T B )-plane and find the region that corresponds to missing the connection.] 3. Show that the function below is the PDF of R , the distance between the epicenter of an earthquake and the site of a dam, when the epicenter is equally likely to be at any location along a neighboring fault (see figure below). You may restrict your attention to a length of fault l that is within a distance r o of the site because earthquakes at greater distances will have negligible effect at the site. f R ( r ) = 2 r ( r 2-d 2 )-1/2 , d ≤ r ≤ r o l Sketch the function. Site r o r o d Fault A B l /2 l /2...
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