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Thread: Schaum's Probability & Random Processes Question 1.11

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    Newbie malweth's Avatar
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    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 $\displaystyle a$ to $\displaystyle b$. The text says:

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

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

    Is the book wrong or do I not understand the intersection and union ($\displaystyle \cap$/$\displaystyle \cup$) correctly?
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    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 $\displaystyle a$ to $\displaystyle b$. The text says:

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

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

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

    -Dan
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    Newbie malweth's Avatar
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    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:

    $\displaystyle a$ -- $\displaystyle s_1$ -- $\displaystyle s_2$ -- $\displaystyle s_3$ -- $\displaystyle b$

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

    Parallel is harder to explain, but essentially:

    $\displaystyle a$ --- $\displaystyle s_1$ --- $\displaystyle b$
    $\displaystyle a$ --- $\displaystyle s_2$ --- $\displaystyle b$
    $\displaystyle a$ --- $\displaystyle s_3$ --- $\displaystyle b$

    (e.g. at least one of $\displaystyle s_1$, $\displaystyle s_2$, or $\displaystyle s_3$ must be down to complete the circuit).
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    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 $\displaystyle a$ to $\displaystyle b$. The text says:

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

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

    Is the book wrong or do I not understand the intersection and union ($\displaystyle \cap$/$\displaystyle \cup$) 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 $\displaystyle (\Omega,\mathcal{F},P).$ $\displaystyle \Omega$ is the space of sample points, here describing the state of the 3 switches. So I would imagine $\displaystyle \Omega$ consists of triples such as (1,0,1) which would say switches 1 and 3 are closed and switch 2 is open. $\displaystyle \mathcal{F}$ are the allowable subsets of $\displaystyle \Omega$, called the events. So the event that switches 1 and 3 are closed and switch 2 is open is the set $\displaystyle A = \{ (1,0,1) \}$. Notice that an event is a set of sample points.

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