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Questions about Linear Combinations of Random Variables

  1. Oct 15, 2013 #1
    1. The problem statement, all variables and given/known data
    RQwRhjc.jpg


    2. Relevant equations
    Y=1/2*(X1-X3)^2+1/14*(X2+2X4-3X5)^2


    3. The attempt at a solution
    For (a) part, I have only learnt to find the moment-generating function of Y, but not finding the p.d.f.
    Moreover, the examples I have seen only involves random variables Xi to the power 1, but not to higher power.

    For (b) part, the difficulty for me is just similar to part (a).
     
    Last edited: Oct 15, 2013
  2. jcsd
  3. Oct 15, 2013 #2
    Hint: [tex] (Z_1^2 + ... +Z_n^2) +(Z_1^2 + ... +Z_m^2) = (Z_1^2 + ... +Z_n^2 +...+ Z_m^2) [/tex]
     
  4. Oct 15, 2013 #3
    sorry but...what property is it? I haven't learnt this before...
     
  5. Oct 15, 2013 #4

    Ray Vickson

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    ##Z_1 = X_1 - X_3## has an ##N(0,a)## distribution, ##Z_2 = X_2+2X_4-3X_5## is ##N(0,b)##, and ##Z_1, Z_2## are independent. So ##Y## involves a weighted sum of squares of independent, mean-0 random variables.
     
  6. Oct 17, 2013 #5
    So do you mean given the random variables are independent, the sum or difference of them will also have the same distribution of them? Also, how can a duel with the square of the random variables? Do I need to consider it with transformation of variables?
     
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