Tools

Random Variables

1. Given two sets F={1,4,6,7} F = \{1, 4, 6, 7 \} and E={0,1,2} E = \{0, 1, 2 \}, what is EΔFE \Delta F?
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2. For a sample space Ω={a,b,c,d}\Omega = \{a,b,c,d\}, consider a collection of subsets C={ϕ,Ω,{a},{b,c,d},{b,c},{a,b},{a,d},{d},{a,b,c}}C = \{ \phi, \Omega, \{a\}, \{b,c,d\}, \{b, c\}, \{a, b\}, \{a, d\}, \{ d\}, \{a, b, c\} \}. If there is one subset which has to be removed from CC such that CC becomes a σ\sigma alebra, which one is it?
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4. For a sample space Ω={w1,w2,w3}\Omega = \{w_1,w_2, w_3\}, and a random variable XX such that X(w1)=−2.5X(w_1) = -2.5 , X(w2)=3X(w_2) = 3 and inverse images X−1(3.5)={w1,w2,w3}X^{-1}(3.5) = \{w_1, w_2, w_3\} and X−1(0)={w1}X^{-1}(0) = \{ w_1\}, what can be the range for the value of X(w3)X(w_3) ?
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