a) LetXbe a real random variable with finite variance. Letmbe the expectation ofX, andv >0 the variance ofX. (a) Show that P(|X-m|=3vv)=1. 9 Give an example of anXwhere equality is attained (that is,...

a) LetXbe a real random variable with finite variance. Letmbe the expectation ofX, andv >0 the variance ofX.

(a) Show that

P(|X-m|=3vv)=1. 9

Give an example of anXwhere equality is attained (that is, the probability of the event above is exactly 1/9).

(b) Let 0=a

variancevsuch that

P(|X-m| =vv/10) =a.

Show that one cannot have

P(|X-m| =vv/10 )= 1.

Remark: in both (a) and (b), it is enough to give thedistributionofXas a probability measure onR.