hw6_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 #6 Due: 11/12/10, in class 1. Consider an experiment where I draw a piece of fruit from a shopping bag. The bag contains an Apple, Banana, Lime, Pear, and Orange. Since I draw them equally likely, I define the probability space as:  ¨£ ¥£ ¡ ©§¦¤¢ 3 ' 1 20)(&%$   ' ¨ £ power set ¥5 64£ 1 ' ! #"  £ ¥   ¡  Apple, Banana, Lime, Pear, Orange The fruit rots at different rates depending on the type. Using advanced science, I calculate how “edible” a fruit is on day to arrive at the sequence of random variables defined by: @8 A9 7  TD £ £  TD £  TD £ e f8 I R VW 7 £  TD H HH R VU 7  TD GHH R SQ 7 HH RX 7   D @ 0FECB8 HH HP 7 R V3 . converge? If so, to what and in what ways? h ‰¦£ 5 D D ˆ#FEC§8   D @ ‚ x @ v£r ¥ wQ ut sqh  c dE  b` FaY @8 g9 (b) Does the probability density function of  (a) Find Apple Banana Lime Pear Orange  ¨£ ¥£ h ©§¦pi 2. An experiment is defined by the probability space , where , is the Borel algebra restricted to , and is defined by . Let . Determine whether or not the sequence converges (and to what) for each of the following cases. Do the parts in order and be sure to justify each answer. ‡„ )26¦†…ƒ(¦&¨   „£ ‚ y €&¨  v£ wQ ut r @8 #9 (a) In distribution. (b) In probability. (c) In quadratic mean. (d) Almost surely. for which: Q ‡ Q  @8 §9 3. Consider the sequence of random variables  t 0‘gBE&¨  @ 8 Q 7  0 Q  @ 8 gBE&¨ 7 @8 §9 ’’’ †††£ W £ U £ Q  7 (b) In probability.  (a) In distribution. t ˆ8  for . Determine whether or not the sequence converges to following cases. Do the parts in order and be sure to justify each answer. for each of the (c) In quadratic mean.  ¨£ ¥£ h ©§¦pi (d) Because the mapping from is not given explicitly, it is not possible to determine whether the sequence converges almost surely to . It could be true or not: t ˆ8  @8 &9 such that: Q ‡ @  t 0‘ Q and definition of   ¨£ ¥£ h ©§¦pi Give an 8 E&¨ 7 Q  0 Q  @ 8 gBE&¨ 7 such that: ‡ @8 &9  t 0‘gBE&¨  @ 8 Q Q7  and definition of . Q ¨©§¦pi £ ¥£ h  @8 g9 Give an does converge almost surely to t ˆ8  for which  0 Q  @ 8 gBE&¨ 7 t ˆ8  does not converge almost surely to converges in mean square to but does not converge with probability , and the definition of the random variables .) £ ¤ (restricted to , of course), and e ¡ , v£ wQ ut r 8 v£ wQ ut r § ¨D @  ¡ ¥ @ § ©D e e ¥ £ ut ¥ ¦t @ § ©D @ @  ¨£ ¥£ h ©#¦pi ¡£ ¡ ¢¤¢ ¨ ©£ be given by , let  D FEC@ @8 &9 ¡ 5 D 5. Let the probability space . For  @8 g9 4. Give an example where one. (Be sure to give . 8  for which ‚ u‡ „ i  „ ‚ ¦†…£ (¦#¨ GHH HH HH HH HH £@ HH HH HH HH £ @ HH HH H I ’’ ††’ HH  FECB8  D @ e HH @   § ©D ¥ @ HH HH £ @ HH HH HH HH HH ’’ ††’ Q ¥ ©D ¥ @ e d@ @ £ e d@ HH HH HP Does this sequence of random variables converge? If so, to what and in what sense (almost surely, in probability, in mean square, in distribution)? ...
View Full Document

This note was uploaded on 09/16/2011 for the course ECE ECE603 taught by Professor Dennisgoeckel during the Fall '10 term at UMass (Amherst).

Ask a homework question - tutors are online