Results 1 to 8 of 8

Math Help - Sum of normal distributions

  1. #1
    Oli
    Oli is offline
    Member
    Joined
    Apr 2008
    Posts
    82

    Sum of normal distributions

    Does anyone know what distribution the sum of two normal distributions will have if they are not independent, but have co-variance a?

    If it is needed, they can have the same mean, but I would rather not have this condition, if possible.

    I have been looking for this, but I can only find it for independent RVs.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Oli View Post
    Does anyone know what distribution the sum of two normal distributions will have if they are not independent, but have co-variance a?

    If it is needed, they can have the same mean, but I would rather not have this condition, if possible.

    I have been looking for this, but I can only find it for independent RVs.
    The sum will be normally distributed. The mean will be the sum of the means,
    and to find the variance:

    V=E([(x1+x2)-(mu1+mu2)]^2)=E( [(x1-mu1)+((x2-mu2)]^2)

    Now expand this and simplify.

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Oli
    Oli is offline
    Member
    Joined
    Apr 2008
    Posts
    82
    Do you know why they will be normally distributed?

    I can see that is how to find the mean and variance if they are.

    Thankyou very much.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Oli
    Oli is offline
    Member
    Joined
    Apr 2008
    Posts
    82
    The question is related to question 4c) of this:
    http://www.maths.ox.ac.uk/filemanager/active?fid=4675

    This is the solution sheet, but in his proof of 4c) he appears to have said that two non-independent normal distribtions sum to give a normal distribution with the variance shown (1). (If you are right about two correlated normal distributions, then this is fine).

    But he has then said by (2) that cov(X,Y)=0 implies independence. I did not think this was true. Any idea why he would say such a thing?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by Oli View Post
    [snip]
    But he has then said by (2) that cov(X,Y)=0 implies independence.[snip]
    This statement is completely wrong. Independence => Cov = 0, but Cov = 0 does NOT => independence.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by Oli View Post
    Does anyone know what distribution the sum of two normal distributions will have if they are not independent, but have co-variance a?

    If it is needed, they can have the same mean, but I would rather not have this condition, if possible.

    I have been looking for this, but I can only find it for independent RVs.
    Oli,

    You have to be careful how you phrase your question.

    If X and Y have a joint normal distribution then their sum has a normal distribution. However, if X and Y each have a normal distribution then it is not necessarily the case that they have a joint normal distribution, and then the result does not follow.

    See the Wikipedia entry on the multivariate normal distribution,
    Multivariate normal distribution - Wikipedia, the free encyclopedia
    especially the section titled "A Counterexample".

    The same web page explains why the sum is normally distributed if X and Y have a joint normal distribution; see the section titled "Affine Transformation".

    It is also true that if X and Y have a joint normal distribution and are uncorrelated then they are independent. The Wikipedia entry says so but does not give a proof. However, a proof is fairly simple. If you look at the joint pdf when the correlation is zero, you will see that it factors into the product of a function of X times a function of Y, which shows that X and Y are independent.

    [Edit]Be careful: If X and Y do not have a joint normal distribution, then being uncorrelated does not imply their independence.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Oli
    Oli is offline
    Member
    Joined
    Apr 2008
    Posts
    82
    Think that sounds exactly right, thanks awkward.
    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 mr fantastic View Post
    This statement is completely wrong. Independence => Cov = 0, but Cov = 0 does NOT => independence.
    But it does for bivariate normals (because the covariance matrix is diagonal
    and so joint distribution comes apart as the product of two normal densities
    in the two variables)

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. normal distributions
    Posted in the Statistics Forum
    Replies: 6
    Last Post: July 19th 2011, 05:50 AM
  2. normal distributions
    Posted in the Statistics Forum
    Replies: 1
    Last Post: October 18th 2010, 07:54 AM
  3. Normal Distributions
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 5th 2010, 06:09 PM
  4. Normal Distributions
    Posted in the Advanced Statistics Forum
    Replies: 18
    Last Post: February 6th 2009, 11:36 AM
  5. How to put two normal distributions together
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: January 17th 2009, 03:33 PM

Search Tags


/mathhelpforum @mathhelpforum