Probabiltity space and random variables

1. Mar 1, 2009

rosh300

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

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

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

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

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

2. Relevant equations
definition of random varibale, probability space

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

(ii) not a random variable

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