Chapter 19 Deviation from the Mean822
Show that for any real number,, and real numbersx > 0, there is anRfor
which the Chebyshev bound is tight, that is,
PrŒjR jxçD
x
2
: (19.26)
Hint:First assumeD 0 and letRtake only the values0; x;andx.
Problem 19.10. (a)A computer program crashes at the end of each hour of use
with probability1=p, if it has not crashed already. IfHis the number of hours
until the first crash, we know
ExŒHçD
1
p
;
VarŒHçD
q
p^2
;
whereqWWD 1 p.
(b)What is the Chebyshev bound on
PrŒjH .1=p/j> x=pç
wherex > 0?
(c)Conclude from part (b) that fora 2 ,
PrŒH > a=pç
1 p
.a 1/^2
Hint:Check thatjH .1=p/j> .a 1/=piffH > a=p.
(d)What actually is
PrŒH > a=pç‹
Conclude that for any fixedp > 0, the probability thatH > a=pis an asymptoti-
cally smaller function ofathan the Chebyshev bound of part (c).
Problem 19.11.
LetRbe a positive integer valued random variable.
(a)If ExŒRçD 2 , how large can VarŒRçbe?
(b)How large can ExŒ1=Rçbe?
(c)IfR 2 , that is, the only values ofRare 1 and 2, how large can VarŒRçbe?