I am reading on Stochastic Dominance (http://en.wikipedia.org/wiki/Stochastic_dominance) and few questions on PDF and CDF.
The paragraph I am looking at this:
Why is that $P[A\ge x] \ge P[B \ge x] $ leads to the cumulative density function $F_A(x) \le F_B(x)$?
Don't understand that part. Need some explanation on that... Thanks..
By definition of the cumulative distribution function, $$ F_A(x) = \mathbb{P}\{A \leq x\} = 1- \mathbb{P}\{A \geq x\} + \mathbb{P}\{A = x\} $$ so for instance when the probability distribution is continuous, $\mathbb{P}\{A = x\}=0$ and you get $$ F_A(x) = 1- \mathbb{P}\{A \geq x\} $$ which implies the relation between the inequalities you point out.