Results 1 to 11 of 11

Math Help - exponential distribution

  1. #1
    Junior Member
    Joined
    May 2010
    Posts
    39

    exponential distribution

    A system with three independent components works correctly if at least one
    component is functioning properly. Failure rates of the individual components are λ1
    = 0.0001, λ2 = 0.0002, and λ3 = 0.0004 (assume exponential lifetime distributions).
    (a) Determine the probability that the system will work for 1000 hours.
    (b) Determine the density function of the lifetime X of the system.

    How do i solve this please help me.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    May 2010
    Posts
    1,030
    Thanks
    28
    This is almost identical to numerous other threads you have posted in the last few weeks. What exactly are you struggling with?

    The CDF of the exponential distribution is:

    P(X\leq x)=1-e^{-\lambda x}
    so
    P(X>x)=e^{-\lambda x}


    So, the probability that the first component is alive after 1000 hours is P(X_1>1000)=e^{-\lambda_1 *1000}
    similarly for X2, and X3 :

    P(X_2>1000)=e^{-\lambda_2 *1000}
    P(X_3>1000)=e^{-\lambda_3 *1000}

    You want P(X1 and X2 and X3 alive after 1000 hours)
    =P(X_1 > 1000) * P(X_2 > 1000) * P(X_3) > 1000
    =e^{-1000\lambda_1 }e^{-1000\lambda_2 }e^{-1000\lambda_3 }
    =e^{-1000(\lambda_1 + \lambda_2 + \lambda_3) }

    Part b
    You want to find F(K) = P(system fails by time K)
    So
    1-F(K) = P(system still alive at time K)


    You can easily find the right hand side by generalising your answer to part a.
    Last edited by SpringFan25; June 18th 2010 at 03:20 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by SpringFan25 View Post
    This is almost identical to numerous other threads you have posted in the last few weeks. What exactly are you struggling with?

    The CDF of the exponential distribution is:

    P(X\leq x)=1-e^{-\lambda x}
    so
    P(X>x)=e^{-\lambda x}


    So, the probability that the first component is alive after 1000 hours is P(X_1>1000)=e^{-\lambda_1 *1000}
    similarly for X2, and X3 :

    P(X_2>1000)=e^{-\lambda_2 *1000}
    P(X_3>1000)=e^{-\lambda_3 *1000}

    You want P(X1 and X2 and X3 alive after 1000 hours)
    For the first part you need the probability P(X1 or X2 or X3 is alive after 1000 hours)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    May 2010
    Posts
    1,030
    Thanks
    28
    true, i should read the question more
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    May 2010
    Posts
    39
    how can i determine the density function
    part b

    please answer
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by ilikec View Post
    how can i determine the density function
    part b

    please answer
    Look at part a) if instead of 1000 hours we do the calculation for $$ x hours to get $$ F(x) then the probability that the system fails between x and x+\delta x hours is F(x)-F(x+\delta x) so in the limit the density of the life time is:

    f(x)=-\dfrac{d}{dx}F(x)

    CB
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    May 2010
    Posts
    39
    captain black isn't this a order statistic sum?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by ilikec View Post
    captain black isn't this a order statistic sum?
    If you have generalised part a) it is irrelevant for part b). The generalised solution to part a) is the probability that the system has not failed by x hours. From that you can calculate the density without further reference to the earlier part of the question.

    CB
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Generalised solution to part a)

    The probability that at least one sub-system is still working at time $$ t is 1 minus the probability that all have failed before $$ t. This is:

     <br />
F(t)=1- (1-e^{-\lambda_1t})(1-e^{-\lambda_2t})(1-e^{-\lambda_3t})<br />

    CB
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Junior Member
    Joined
    May 2010
    Posts
    39
    captain black your generalized solution is for 1st order statistic. Right?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by ilikec View Post
    captain black your generalized solution is for 1st order statistic. Right?
    Why are you concerned about the name rather than is it right? It is (if I have not made a mistake) the probability that the system is still operational at time t, which is the probability that one or more sub-systems is still functional at t.

    Also I am not that familiar with order statistics and would have to look them up if I were to try to answer that question, which you can do as easily as me

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Exponential distribution
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 14th 2010, 04:08 PM
  2. Replies: 0
    Last Post: March 14th 2010, 06:49 AM
  3. PDF exponential distribution
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: February 14th 2009, 08:34 PM
  4. exponential distribution
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 2nd 2008, 01:10 PM
  5. Exponential distribution
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: March 22nd 2008, 03:35 PM

Search Tags


/mathhelpforum @mathhelpforum