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Math Help - Adding Normal distributions

  1. #1
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    Adding Normal distributions

    How do I derive the poisson distribution?
    Last edited by bluebiro; September 4th 2009 at 12:22 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by bluebiro View Post
    Say D is distributed by N(2,2), E is distributd by N(3,3) and F is distributed by N(4,4), and D, E and F are independent.

    (a) Find E(D squared plus E squared plus F squared)
    I get 2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29
    By linearity of the expectation operator:

    E(D^2+E^2+F^2)=E(D^2)+E(E^2)+E(F^2)

    Now:

    var(X)=E(X^2)-\overline{X}^2

    so:

    E(X^2)=var(X)+\overline{X}^2

    CB
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    Quote Originally Posted by bluebiro View Post
    Say D is distributed by N(2,2), E is distributd by N(3,3) and F is distributed by N(4,4), and D, E and F are independent.
    (b) Find Cov(D+E, D-2E)
    (stuck)
    Cov(D+E, D-2E)=E( [(D+E)-(\overline{D}+\overline{E})][(D-2E)-(\overline{D}-2\overline{E})]

    Now expand the expression inside the expectation on the right hand side. Then use linearity of the expectation operator to express the right hand side as a sum of expectations, and use the fact that the means and variances of all the RV's are known and that they are independent to expand the resulting expression.

    CB
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    Quote Originally Posted by bluebiro View Post
    Say D is distributed by N(2,2), E is distributd by N(3,3) and F is distributed by N(4,4), and D, E and F are independent.

    (c) What is the resulting Normal distribution of D+E+F?
    I get N(0, 6)
    How do you get that, if you explain we can see what the problem is with your understanding and so help you.

    CB
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  5. #5
    MHF Contributor matheagle's Avatar
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    (b) Cov(D+E, D-2E)=Cov(D,D)+Cov(E,D)-2Cov(D,E)-2Cov(E,E)=V(D)+0+0-2V(E)
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    Quote Originally Posted by CaptainBlack View Post
    How do you get that, if you explain we can see what the problem is with your understanding and so help you.

    CB
    Sorry, I don't know what I was doing there.

    I get (2,12) as the new distribution this time?
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    Quote Originally Posted by bluebiro View Post
    Sorry, I don't know what I was doing there.

    I get (2,12) as the new distribution this time?
    My question is: What are you doing to get this answer (if you are guessing tell us you have no idea what to do)?

    CB
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    Quote Originally Posted by bluebiro View Post
    Sorry, I don't know what I was doing there.

    I get (2,12) as the new distribution this time?
    Read this: Sum of normally distributed random variables - Wikipedia, the free encyclopedia
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    Quote Originally Posted by CaptainBlack View Post
    My question is: What are you doing to get this answer (if you are guessing tell us you have no idea what to do)?

    CB
    Sorry, my original question was typed wrong. I wanted to find the resulting distribution of D+E-F...
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    Quote Originally Posted by bluebiro View Post
    Sorry, my original question was typed wrong. I wanted to find the resulting distribution of D+E-F...
    and my working was:

    E(new function)=3+4-5=2
    Var(new function)=3+4+5=12
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    Quote Originally Posted by bluebiro View Post
    and my working was:

    E(new function)=3+4-5=2
    Var(new function)=3+4+5=12
    Which looks OK.

    CB
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  12. #12
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    Quote Originally Posted by matheagle View Post
    (b) Cov(D+E, D-2E)=Cov(D,D)+Cov(E,D)-2Cov(D,E)-2Cov(E,E)=V(D)+0+0-2V(E)
    Why is the covariance (E,D) = 0 ?
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    Quote Originally Posted by bluebiro View Post
    Why is the covariance (E,D) = 0 ?
    E and D are independent RV's and:

    Covar(E,D)=E((E-E(E))(D-E(D)))=E(E-E(E)) E(D-E(D))

    and (E-E(E)) and (D-E(D)) are zero mean RV's.

    CB
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    Quote Originally Posted by CaptainBlack View Post
    E and D are independent RV's and:

    Covar(E,D)=E((E-E(E))(D-E(D)))=E(E-E(E)) E(D-E(D))

    and (E-E(E)) and (D-E(D)) are zero mean RV's.

    CB
    (sorry for all the questions)

    What does "zero mean RV's" mean?
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  15. #15
    Grand Panjandrum
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    Quote Originally Posted by bluebiro View Post
    (sorry for all the questions)

    What does "zero mean RV's" mean?
    RV :- Random Variable

    "zero mean":- has a mean of zero.

    CB
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