Results 1 to 8 of 8

Math Help - find distribution

  1. #1
    Junior Member
    Joined
    May 2010
    Posts
    39

    find distribution

    suppose that a certain device consists of 10 components and each component's life time can be modeled using a exponential distribution with mean 100 hours.

    a) assuming that the device becomes out of order when one component burns. Find the distribution functionn for the lifetimee of this device.

    b) assume that the device will be put to work even if at least one component is working fund the distribution functionn for the lifetime for this device.

    I've got no idea to proceed with this problem. Please help me.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    May 2010
    Posts
    1,028
    Thanks
    28
    part a

    Quick method:
    as long as the failure rates of the individual components are independant, the time to first failure is distributed as exponential with mean 10.

    You can show this rigorously by noting that the exponential distribution is the waiting time of a poisson process, and that the sum of iid poisson variables is also poisson with the parameters added up.


    Long Method:
    Let T be the time to failure of the System
    P(T<k) = P(at least 1 component dead by k)
    P(T<k) = 1-P(all components alive at k)
    P(T<k) = 1-(P(component 1 alive at k)*P(component 2 alive at k).....)

    You can easily work out the chance of each component being alive at K, since you were given the distribution for each component.








    part (b).
    let T be the failure time of the system

    P(T<k) = P(all components dead at time k)
    P(T<k) = P(first component dead at time k) * P(second component dead at time k) * ...etc

    Each term on the right hand side is the CDF of 1 component (which you know). Multiply them together to get your answer
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by ilikec View Post
    suppose that a certain device consists of 10 components and each component's life time can be modeled using a exponential distribution with mean 100 hours.

    a) assuming that the device becomes out of order when one component burns. Find the distribution functionn for the lifetimee of this device.

    b) assume that the device will be put to work even if at least one component is working fund the distribution functionn for the lifetime for this device.

    I've got no idea to proceed with this problem. Please help me.
    For (a) I think you need to consider the distribution of min(X1, X2, ... X10).

    For (b) I think you need to consider the distribution of max(X1, X2, ... X10).

    Both are simple order statistics problems and you will find the required distributions easily using Google.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    May 2010
    Posts
    39
    what does find distribution function mean?

    Please help.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    May 2010
    Posts
    1,028
    Thanks
    28
    the distribution function gives the probability that X is less than or equal to some value.

    <br />
F(x) = P(X \leq x)<br />
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    May 2010
    Posts
    39
    Can someone please give me links to online resources that explains well about order statistics.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    May 2010
    Posts
    1,028
    Thanks
    28
    You could try Order statistic - Wikipedia, the free encyclopedia

    But, if you have not heard of order statistics before then you dont need to learn it to solve this problem. You can use the reasoning i put in my earlier reply.


    Here is part (a), step by step, To get you started.

    Concept - the distribution function
    The distribution function of a random variable is normally written F(x). It tells you the probability that the random variable X takes a value less than or equal to some particular value (x).
    F(x) = P(X \leq x)


    Define:
    X_1 .... X_{10} the lifetime of the 10 components
    F_i(x) = P(X_i \leq x) the distribution function for each X
    T the lifetime of the system
    G(t) = P(T \leq t) the distribution function for T


    Information from question
    X_i are independant and follow the same distribution: exp(\lambda)
    We are told the mean value of each distribution is 100, and you should know (or be able to look up) that the mean value of an exponential distribution is 1/\lambda
    so X_i \sim exp(0.01)

    You should know (or be able to look up) that the distribution function for he exponential distribution is
    F_i(x) = 1-e^{-\lambda x} = 1-e^{-0.01x}




    Question
    You are told that the system fails when 1 component fails. Find G(t).

    Step 1
    G(t) = P(T<t)

    What is the probability of the system fialing before time t? This happens if at least one component has failed by time t.
    G(t) = P(T<t) =P(at least 1 component failes by time t)
    G(t) = P(T<t) =1-P(no~component~fails~by~time~t)


    Step 2
    What is the probability that no component has failed by time t?
    P(no failures) = P(X_1~alive~and X_2~alive~and~X_3~alive~and...)

    P(no failures) = P(X_1~alive)P(X_2~alive)P(X_3~alive)P(X_4~alive)..  .

    P(no failures) = P(X_1 > t)P(X_2 > t)P(X_3 > t)P(X_4>t)...

    P(no failures) = \left(1-F_1(t) \right) \left(1-F_2(t) \right) \left(1-F_3(t) \right) \left(1-F_4(t) \right) \left(1-F_5(t)]\right)...

    P(no failures) = [1-F(t)]^{10} (all X's follow the same distribution)

    P(no failures) = [1-F(t)]^{10}

    P(no failures) = [1-1-e^{-0.01t}]^{10}

    P(no failures) = [e^{-0.01t}]^{10} = e^{-0.1t}

    Step 3
    Put it all together:

    G(t) = P(T<t) =1-P(no~failures~fails~by~time~t)
    G(t) = 1-e^{-0.1t}

    You can recognise this as an exponential(0.1) distribution.

    Part a Answer
    The lifetime of the system follows an exp(0.1) distribution.
    Last edited by SpringFan25; June 26th 2010 at 09:38 AM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by ilikec View Post
    Can someone please give me links to online resources that explains well about order statistics.
    Google does a good job of doing that.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Find the distribution
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: November 23rd 2009, 11:07 AM
  2. Find the distribution
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: November 2nd 2009, 02:41 PM
  3. find mle of this distribution
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: October 20th 2009, 02:39 PM
  4. How to find out distribution's mode?
    Posted in the Statistics Forum
    Replies: 2
    Last Post: June 4th 2009, 09:36 PM
  5. [SOLVED] How To Find The Distribution
    Posted in the Algebra Forum
    Replies: 1
    Last Post: April 25th 2005, 03:33 AM

Search Tags


/mathhelpforum @mathhelpforum