hw4_fall10 - ECE 603 - Probability and Random Processes,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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  ¨¦ 3a`§¤ ¢ ¡ 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  ¨¦ 3a©§¤ ¢ ¡ !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 T@ R A SQf ) 3 R@ A SQf % R@ 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 ...
View Full Document

Ask a homework question - tutors are online