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Probabiltity space and random variables

  1. Mar 1, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]\Omega [/tex] is a set of points [tex]\omega ; C_{i} i [/tex] = 1, 2, ... 7 are subsets of [tex] \Omega[/tex];
    and ([tex] \Omega[/tex], F, P) = ([tex]B_{i}, i/10, i = 1, 2, 3, 4 [/tex]) is a probability modal
    with [tex] B_{1} = C_{1} \cup C_{7}, B_{2} = C_{2} \cup C_{6}, B_{3} = C_{3} \cup C_{5} and B_{4} = C_{4}[/tex].
    State which of the following functions X:[tex]\Omega \rightarrow [/tex] R are random variables defined on (\Omega, F, P) and derive the distributions.

    (i)[tex] X(\omega) = -3 [/tex] for [tex] \omega \in C_{1} \cup C_{7} \cup C_{3} \cup C_{5} [/tex] with [tex] X(\omega) = 2 [/tex] otherwise

    (ii) [tex]X(\omega) = 1 for \omega \in C_{1} \cup C_{7}, X(\omega) = 2 [/tex] for [tex] \omega \in C_{3} \cup C_{4} and X(\omega) = 2 [/tex] for [tex] \omega \in C_{2} \cup C_{5} \cup C_{6} [/tex]

    (iii)[tex] X(\omega) = (v-4)^{2} for \omega \in C_{v}, v = 1, 2, ... 7 [/tex]

    2. Relevant equations
    definition of random varibale, probability space


    3. The attempt at a solution
    (i) random variable
    Distrubution:
    (-[tex]\infty [/tex], -3) = 0
    [-3, 2) = 1/10 + 3/10 = 2/5
    [2,[tex] \infty [/tex]) = 1

    (ii) not a random variable

    (iii) random variable
    distrubution:
    (-[tex]\infty [/tex], 0) = 0,
    [0, 1) = 4/10
    [1, 4) = 7/10
    [4, 9) = 9/10
    [9, [tex]\infty [/tex]] = 1
     
  2. jcsd
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