Positive and negative parts
Let X take the values -2, 0, and 3 with probabilities 1/4, 1/2, and 1/4. Find F_X, \mathbb{E} X^+, \mathbb{E} X^-, and \mathbb{E} X.
Worked answer (click to show)
The distribution function is 0 for x<-2, equals 1/4 for -2\le x<0, equals 3/4 for 0\le x<3, and equals 1 for x\ge 3. The positive part is 3 with probability 1/4 and 0 otherwise, so \mathbb{E} X^+=3/4. The negative part is 2 with probability 1/4 and 0 otherwise, so \mathbb{E} X^-=1/2. Therefore \mathbb{E} X=3/4-1/2=1/4. The direct sum -2\cdot(1/4)+0\cdot(1/2)+3\cdot(1/4)=1/4 agrees.