Results 1 to 4 of 4

Math Help - Schaum's Probability & Random Processes Question 1.11

  1. #1
    Newbie malweth's Avatar
    Joined
    Jun 2007
    From
    Coventry, RI
    Posts
    4

    Schaum's Probability & Random Processes Question 1.11

    I was going to post this yesterday, but thought I understood it ... and now I don't:

    In Schaum's Probability, Random Variables, & Random Processes by Hwei Hsu, problem 1.11 gives four switched circuit diagrams, each with 3 switches. The goal is to define the path from a to b. The text says:

    Consider the switching networks shown in Fig. 1-5. Let A_1, A_2, and A_3 denote the events that the switches s_1, s_2, and s_3 are closed, respectively. Let A_{ab} denote the event that there is a closed path between terminals a and b. Express A_{ab} in terms of A_1, A_2, and A_3 for each of the networks shown.

    The two networks I'm concerned with are (a) and (b), which simply have the switches in series and parallel, respectively. The answers given are:

    (a) A_{ab} = A_1 \cap A_2 \cap A_3
    (b) A_{ab} = A_1 \cup A_2 \cup A_3

    This doesn't make sense to me. It is true that in (a), all switches must be switched, and I would agree that A_{ab} = \{A_1A_2A_3\} (in fact, that was my answer to the problem) but I don't see how \{A_1A_2A_3\} = A_1 \cap A_2 \cap A_3 unless A_1 = A_2 = A_3. Problem (b) has a similar issue.

    Is the book wrong or do I not understand the intersection and union ( \cap/ \cup) correctly?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,937
    Thanks
    337
    Awards
    1
    Quote Originally Posted by malweth View Post
    I was going to post this yesterday, but thought I understood it ... and now I don't:

    In Schaum's Probability, Random Variables, & Random Processes by Hwei Hsu, problem 1.11 gives four switched circuit diagrams, each with 3 switches. The goal is to define the path from a to b. The text says:

    Consider the switching networks shown in Fig. 1-5. Let A_1, A_2, and A_3 denote the events that the switches s_1, s_2, and s_3 are closed, respectively. Let A_{ab} denote the event that there is a closed path between terminals a and b. Express A_{ab} in terms of A_1, A_2, and A_3 for each of the networks shown.

    The two networks I'm concerned with are (a) and (b), which simply have the switches in series and parallel, respectively. The answers given are:

    (a) A_{ab} = A_1 \cap A_2 \cap A_3
    (b) A_{ab} = A_1 \cup A_2 \cup A_3

    This doesn't make sense to me. It is true that in (a), all switches must be switched, and I would agree that A_{ab} = \{A_1A_2A_3\} (in fact, that was my answer to the problem) but I don't see how \{A_1A_2A_3\} = A_1 \cap A_2 \cap A_3 unless A_1 = A_2 = A_3. Problem (b) has a similar issue.

    Is the book wrong or do I not understand the intersection and union ( \cap/ \cup) correctly?
    Where's Figure 1-5!?

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie malweth's Avatar
    Joined
    Jun 2007
    From
    Coventry, RI
    Posts
    4
    The two networks I'm concerned with are (a) and (b), which simply have the switches in series and parallel, respectively.
    I wouldn't know how to draw it.

    Series means that you have:

    a -- s_1 -- s_2 -- s_3 -- b

    (e.g. all three switches must be down to connect a to b).

    Parallel is harder to explain, but essentially:

    a --- s_1 --- b
    a --- s_2 --- b
    a --- s_3 --- b

    (e.g. at least one of s_1, s_2, or s_3 must be down to complete the circuit).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Apr 2006
    Posts
    399
    Awards
    1
    Quote Originally Posted by malweth View Post
    I was going to post this yesterday, but thought I understood it ... and now I don't:

    In Schaum's Probability, Random Variables, & Random Processes by Hwei Hsu, problem 1.11 gives four switched circuit diagrams, each with 3 switches. The goal is to define the path from a to b. The text says:

    Consider the switching networks shown in Fig. 1-5. Let A_1, A_2, and A_3 denote the events that the switches s_1, s_2, and s_3 are closed, respectively. Let A_{ab} denote the event that there is a closed path between terminals a and b. Express A_{ab} in terms of A_1, A_2, and A_3 for each of the networks shown.

    The two networks I'm concerned with are (a) and (b), which simply have the switches in series and parallel, respectively. The answers given are:

    (a) A_{ab} = A_1 \cap A_2 \cap A_3
    (b) A_{ab} = A_1 \cup A_2 \cup A_3

    This doesn't make sense to me. It is true that in (a), all switches must be switched, and I would agree that A_{ab} = \{A_1A_2A_3\} (in fact, that was my answer to the problem) but I don't see how \{A_1A_2A_3\} = A_1 \cap A_2 \cap A_3 unless A_1 = A_2 = A_3. Problem (b) has a similar issue.

    Is the book wrong or do I not understand the intersection and union ( \cap/ \cup) correctly?
    It seems to me you are not understanding the difference between events and sample points in a probability space.

    A probability space is a triple (\Omega,\mathcal{F},P). \Omega is the space of sample points, here describing the state of the 3 switches. So I would imagine \Omega consists of triples such as (1,0,1) which would say switches 1 and 3 are closed and switch 2 is open. \mathcal{F} are the allowable subsets of \Omega, called the events. So the event that switches 1 and 3 are closed and switch 2 is open is the set A = \{ (1,0,1) \}. Notice that an event is a set of sample points.

    The event that switch s_i is closed would be A_i = \{ (s_1,s_2,s_3) \in \Omega \ | \ s_i = 1 \ \}. The event A that switches 1 and 3 are closed and switch 2 is open can be written as A = \{ (1,0,1) \} = A_1 \cap A_2^C \cap A_3. The event A_{ab} switches 1, 2 and 3 are closed is A_{ab} = \{ (1,1,1) \} = A_1 \cap A_2 \cap A_3 as the book says. The event A_{ab} that at least one of switches 1, 2 and 3 are closed is A_{ab} = \{ (s_1,s_2,s_3) \in \Omega \ | \ s_1 = 1 \vee s_2 = 1 \vee s_3 = 1 \} = A_1 \cup A_2 \cup A_3.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Comparison of stochastic processes at a random time 2
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: August 3rd 2010, 07:52 AM
  2. Comparison of stochastic processes at a random time
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: August 1st 2010, 05:32 AM
  3. Probability and random random varible question
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 15th 2010, 08:53 PM
  4. Random Processes (Martingale)
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: April 25th 2010, 09:47 PM
  5. Markov Processes and Random Walks
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: October 6th 2008, 10:39 AM

Search Tags


/mathhelpforum @mathhelpforum