hw4_fall10

# hw4_fall10 - ECE 603 Probability and Random Processes Fall...

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2010 Homework #4 Due: 10/15/10, in class has probability density function: %# &\$¦ 1. An exponential random variable with parameter !% !  "  else   ¨ ¦¤ ¢ ©§¥£¡ Suppose that I have three lightbulbs of varying qualities: . D E¤ C  ¨R¦ Y£XW# B DT R ¨ I G ¦ ¤ VUSQPH§F# B D E¤ C 9 6( ' ' 9 @( is said to be memoryless if . B is an exponential random variable with . # is an exponential random variable with A 8 4 5( hours, where 4 6( ) 0( B Deﬁnition: A random variable for all and . hours, where is an exponential random variable with 7 8 ' The third dies out after ) 1( The second dies out after hours, where 2 3 The ﬁrst dies out after I ¦ ¨R £EI (a) Suppose that I use the ﬁrst lightbulb (which lasts a random time ) in my lamp, and it has lasted two hours. Given such, what is the chance it lasts more than four hours? )( ) 1( (b) Is memoryless? (c) I pick a lightbulb at random (i.e. each of the three is equally likely) and put it into my lamp. Denote the time that the lamp is lit as . What is the probability that the lamp is lit for more than 2 hours? ( (d) I pick a lightbulb at random (i.e. each of the three is equally likely) and put it into my lamp. Denote the time that the lamp is lit as . The lamp is lit for two hours. Given such, what is the probability that it is lit for more than four hours? ( memoryless? Explain any difference from (b). b  ¨¦ 3a`§¤ ¢ ¡ b . #4 B . 4 r 7 WG B t uG 4 B qi p A (d) Find ) rp sB qi (c) Find the probability that . !% (b) Find is given by 2 f¦f2c T FhWged¦ T (a) Find the value of the constant . B 2. The probability density function of a random variable otherwise ¨ ¦¤ ¢ `§¥"¡ ( (e) Is , where: 3. Suppose that your goal is to maximize the proﬁt of your business. If you decide to travel to Xville, the proﬁt (in dollars) for your business is a random variable with cumulative distribution function as given below. If you decide to travel to Yville, the proﬁt (in dollars) for the trip is a random variable with cumulative distribution function as given below. B ¨ ¥¤ ¤ ¨ ¥¤ ¤ ¨¦ `§¤ ¢ ¢ £ ¡ ¢ ¦ ¨¦ ©§¤ ¢ 1 3 1 3 2 ¤ 2 ¨4 1 ¦ -1 ¤¤ 1 (a) To which village would you travel to maximize your proﬁt? ¨ £ . Find the cumulative distribution function . : B  § 2 % # !"  &\$ (  ' ¨ ¦¤ ©§@ is given by ¨¦ `§¤ ¢ ¡ 4    ¨ ¦¤ Y©§@ 7 . Write an , where: 2 B ¤ ¤ 2f ¨ ¡ ¤ ¡ C 5. The probability density function of a random variable ; that is, ( ' 0¡ 1) G4 ¡ Let be a Gaussian random variable with mean and variance expression for for all in terms of . ! 4 4. You have a table that gives you the value of the “Goeckel Function” for all %# &h¦ ¨ ¤ © (b) Let f Wf 0c ¦ 7 !% b  ¨¦ 3a©§¤ ¢ ¡ !4¦ otherwise (a) Find the value of the constant . b #2 g&c B . be deﬁned by: 2!% B B c B (where the Borel ﬁeld is restricted 2Ff 2Ff )3 f 8 9 6 r A R ) 3 [email protected] R A SQf ) 3 [email protected] A SQf % [email protected] p 4    !% ¡ ¡ ¡ ¨ C !6 q7! 4 5¤ 4 4 IP H 4 G ) EG IP4  H DY¨ ¨ B! ¤ ¤ CE F  A @ I P H r ! ! . % &# ! 2!% p C rp sB qi . Find , . 6. Let the probability space be given by to , of course) and, for any interval, 4 U ¨U 9a6¥¤ B Let f Find the probability density function of % (d) Let the random variable . )3 c B (c) Find the probability that 2 F# 4 (b) Find the probability that ...
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