Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.10. Important Inequalities 83

1.10.2.LetXbe a random variable such thatP(X≤0) = 0 and letμ=E(X)
exist. Show thatP(X≥ 2 μ)≤^12.

1.10.3. IfX is a random variable such thatE(X)=3andE(X^2 ) = 13, use
Chebyshev’s inequality to determine a lower bound for the probabilityP(− 2 <
X<8).

1.10.4.SupposeX has a Laplace distribution with pdf (1.9.20). Show that the
mean and variance ofXare 0 and 2, respectively. Using Chebyshev’s inequality
determine the upper bound forP(|X|≥5) and then compare it with the exact
probability.

1.10.5.LetXbe a random variable with mgfM(t),−h<t<h.Provethat

P(X≥a)≤e−atM(t), 0 <t<h,

and that
P(X≤a)≤e−atM(t), −h<t< 0.
Hint:Letu(x)=etxandc=etain Theorem 1.10.2. Note: These results imply
thatP(X≥a)andP(X≤a) are less than or equal to their respective least upper
bounds fore−atM(t)when0<t<hand when−h<t<0.

1.10.6.The mgf ofXexists for all real values oftand is given by

M(t)=

et−e−t

2 t

,t =0,M(0) = 1.

Use the results of the preceding exercise to show thatP(X≥1) = 0 andP(X≤
−1) = 0. Note that herehis infinite.

1.10.7.LetXbe a positive random variable; i.e.,P(X≤0) = 0. Argue that

(a)E(1/X)≥ 1 /E(X)

(b)E[−logX]≥−log[E(X)]

(c)E[log(1/X)]≥log[1/E(X)]

(d)E[X^3 ]≥[E(X)]^3.