Tools

Random Variables

1. For a sample space Ω={w1,w2,w3,w4}\Omega = \{w_1,w_2,w_3,w_4\}, and a function X:Ω←RX:\Omega \leftarrow \mathbb{R} such that X(w1)=X(w3)=X(w4)=2.5X(w_1) = X(w_3) = X(w_4) = 2.5 and X(w2)=1X(w_2) = 1 . What is the smallest possible σ\sigma algebra F\mathcal{F} that makes XX a random variable?
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2. Given a Ω={1,2,3,4}\Omega = \{1, 2, 3, 4\} and F={ϕ,Ω,{1},{2},{1,2},{3,4},{1,3,4},{2,3,4}}\mathcal{F} = \{ \phi, \Omega, \{1\}, \{2\}, \{1,2\}, \{3,4\}, \{1,3,4\}, \{2,3,4\}\} and a valid random mapping X(1)=5,X(2)=4,X(3)=1,X(4)=1X(1) = 5, X(2) = 4, X(3) = 1, X(4) = 1 and the inverse image of cc, denoted by X−1(c)={2,3,4}X^{-1}(c) = \{ 2, 3, 4\} what is the valid range for values of cc
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3. For a sample space Ω={1,2,3,4}\Omega = \{1, 2, 3, 4\} and an event space F={ϕ,Ω,{1},{2},{1,2},{3,4},{1,3,4},{2,3,4}}\mathcal{F} = \{ \phi, \Omega, \{1\}, \{2\}, \{1,2\}, \{3,4\}, \{1,3,4\}, \{2,3,4\}\}, which of the following is a valid random variable mapping?
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4. Let the sample space Ω\Omega = {1,2,3,4} and a random variable Y defined as Y(1)=1,Y(2)=1,Y(3)=2,Y(4)=2Y(1)=1, Y(2)=1, Y(3)=2, Y(4)=2. What is the set Y−1(1)Y^{-1}(1)?
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