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Help with probability distribution function question

  1. Jul 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Let X and Y be two independent random variables with the same probability density funtion over:

    f(x) = {1/a if x € [0,a]
    {0 if x=0

    Find the density distribution of a) X + Y and b) X*Y


    2. Relevant equations



    3. The attempt at a solution

    Ok, my initial thoughts are for:

    a) That f(X+Y) is simply the product of each of their densities, which would result in 1/(a^2)

    b) f(X*Y) is the sum of their individual distributions. i.e. 2/a


    This seems a little too simple for me and I think I am looking at it from the wrong perspective.

    Any thoughts or advice?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 28, 2011 #2
    No, i don't think this is right. Just consider that the two random variables are idependently and identically distributed w.r.t the distribution you mentioned above. So [itex] X + Y [/itex]takes its values in the interval [itex] [0, 2a][/itex]. What you were referring to is the Gaussian distribution where the result concerning a would be valid. In this case however, since the points are distributed according to (1) a continous distribution which is (2) uniform, we conclude that if the two random variable [itex]X, Y[/itex] also are distributed according to this distribution, then there sum has to be distributed (again uniformly) with probability density [itex] 1/2a [/itex]. Converesely, take question b. You may regard X and Y as some sort of random coordinates which give you point lying a sqaure with side length a. Again, recalling that we deal with uniform distributions, we find the probability density function to be constant [itex] 1/a^{2}[/itex]. The problem is that one has to distinguish between uniform distributions and the standard distributions, i.e., Gaussian, binomial, geometric, multinomial, Poisson, exponential.
     
  4. Jul 28, 2011 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    You can get the density f(z) of X+Y by differentiating the CDF F(z) = Pr{X+Y <= z}, and the latter is an integral over the two-dimensional region {x+y <= z, 0 <= x,y <= a}. Or, Google "convolution".

    RGV
     
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