PROBLEM SET 8
1. Let ? and @ be continuous random variables having a bivariate normal distribution
with means ? and @ , common variance , and correlation coefficient ?@ . Let -?
and -@ be the cumulative distribution functions of ? and @ respectively. Determine
which of the following is a necessary and sufficient condition for -? ²!³ ‚ -@ ²!³ for all !.
A) ? ‚ @
B) ?  @
C) ? ‚ ?@ @
D) ?  ?@ @
E) ?@ ‚ 2. Let ? and @ be discrete random variables with joint probability function
²%Á &³ ~ H
A) %bc&
for %~Á and &~Á
Á otherwise
B) C) D) . Calculate ,´ ?
@ µ.
E) ~ Á
3. Let ? and @ be independent random variables with ? ~ Á @ ~ c Á ?
and @ ~ . Calculate ,´²? b ³ ²@ c ³ µ .
A) B) C) D) E) 4. Let ? and @ be continuous random variables with joint density function
²%Á &³ ~ F
for %&
, otherwise
.
Determine the density function of the conditional distribution of @ given ? ~ %, where
 %  .
for %  &  A) %c
B) ² c %³ for %  &  C) for %  &  D) & for %  &  for %  &  E) c&
5. Let ? and @ be continuous random variables with joint cumulative distribution
²%& c % & c %& ³ for  %  and  &  .
function - ²%Á &³ ~ Determine 7 ´? € µ .
²& c & ³
²
& c & ³
A) B) C) D) E) 6. Let ? and @ be discrete random variables with joint probability function ²%Á &³ given
by the following table:
%
À
À
À
À
&
À
À
À
À
For this joint distribution, ,´?µ ~ À and ,´@ µ ~ . Calculate *#´?Á @ µ .
A) c À
B) c À
C) À
D) À
E) À
7. A wheel is spun with the numbers 1, 2 and 3 appearing with equal probability of each. If the number 1 appears, the player gets a score of 1.0; if the number 2 appears, the
player gets a score of 2.0; if the number 3 appears, the player gets a score of ? , where ?
is a normal random variable with mean 3 and standard deviation 1. IF > represents the
player's score on 1 spin of the wheel, then what is 7 ´>  Àµ?
A) À
B) À
C) À
D) À
E) À
8. Let ? and @ be discrete random variables with joint probability function
²%Á &³ ~ F
&
%
for %~ÁÁ  &~ÁÁ  %&
Á otherwise
What is 7 ´? b @  µ?
A) B) C) .
D) E) 9. Let ? and @ be continuous random variables with joint density function
²%Á &³ ~ F
À% for % and &c%
Á otherwise
What is 7 ´? € µ ?
A) B) C) D) .
E) 10. Let ? and @ be continuous random variables with joint density function
²%Á &³ ~ F
for &cO%O and c%
Á otherwise
What is = ´?µ ?
A) B) C) D) .
E) 11. Let ? and @ be continuous random variables with joint density function
²%Á &³ ~ F
%b& for % Á &
Á otherwise
.
What is the marginal density function for ?Á where nonzero?
A) & b B) %
D) %b%
C) %
E) % b 12. Let ? be a random variable with mean 3 and variance 2, and let @ be a random
variable such that for every %, the conditional distribution of @ given ? ~ % has a mean
of % and a variance of % . What is the variance of the marginal distribution of @ ?
A) B) C) D) E) 13. Let ? and @ be discrete random variables with joint probability function
²%Á &³ ~ F
²%b³²&b³
for %~ÁÁ Â &~ÁÁ
Á otherwise
What is ,´@ O? ~ µ?
A) B) C) D)
&b
.
E)
& b&
14. Let ? and @ be continuous random variables with joint density function
²%Á &³ ~ F
% for %&
.
Á otherwise
Note that ,´?µ ~ and ,´@ µ ~ . What is *#´?Á @ µ ?
A) B) C) D) E) 15. Let ? and @ be discrete random variables with joint probabilities given by
?
b b @
b b Let the parameters and satisfy the usual assumption associated with a joint
probability distribution and the additional constraints c À   À and
  À . If ? and @ are independent, then ² Á ³ ~
Á ³
A) ²Á ³ B) ² Á ³ C) ² c Á ³ D) ² c Á ³ E) ² 16. Let ? and @ be continuous random variables with joint density function
²%Á &³ ~ F
A) %& for % and &
Á otherwise
B) C) What is 7 ´ ?
 @  ?µ ?
.
D) E) 17. Let @ have a uniform distribution on the interval ²Á ³, and let the conditional
distribution of ? given @ ~ & be uniform on the interval ²Á l&³. What is the marginal
density function of ? for  %  ?
A) ² c %³
B) %
C) ² c %° ³
D) % c E) %
l
18. Let ²?Á @ ³ have joint density function ²%Á &³ ~ F
For  %  , what is = ´@ O? ~ %µ ?
A) ²c%³
l
for %&
Á otherwise
.
c%
D) ²c%³
E) Cannot be determined from the given information
B)
C) b%
19. If the joint probability density function of ? Á ? is ²% Á % ³ ~ , for
 %  and  %  , and otherwise, then the moment generating function
4 ²! Á ! ³ Á ! Á ! £ of the joint distribution is
!
A) !c
B)
(! c)(! c)
! !
C)
(! b)(! b)
! !
D) ! !
E) ! b! c 20. The moment generating function for the joint distribution of random variables ? and
!
@ is 4 ²! Á ! ³ ~ ²c!
³ b h ²c! ³ , for !  . Find = ´?µ .
A) B) C) D) E) 21. Let ? Á ? Á ? be uniform random variables on the interval ²Á ³ with
for Á ~ Á Á Á £ . Calculate the variance of ? b ? c ? .
*#´? Á ? µ ~ A) B) C) D) E) 22. Let ? Á ? Á ? be independent discrete random variables, with the probability
function 7 ´? ~ µ ~ 4 5 ² c ³ c for ~ Á Á ÀÀÀÁ for ~ Á Á and   .
Determine the probability function of : ~ ? b ? b ? Á 7 ´: ~ µ.
A) 6
b b
7 ² c ³ b b c B) b b 4 5 ² c ³ c
~
C) 4 5 ² c ³ c
D) 4 5 ² c ³ c
~
E) 4 5 ² c ³ c
~
23. Let ? be a continuous random variable with density function
²%³ ~ F
%c for %‚
Á otherwise
.
for  &  .
Determine the density function of @ ~ ?c
&
&b
A) &
B) ²&b³
C) ²&c³
D) ² &b ³
E) ² & ³
24. Let ? and @ be two independent random variables with moment generating
functions
4? ²!³ ~ ! b! , 4@ ²!³ ~ ! b!
Determine the moment generating function of ? b @ .
A) ! b! b ! b!
B) ! b! b ! b! C) 7! b!
D) ! b!
E) ! b!
25. Let ? and ? be random variables with joint moment generating function
4 ²! Á ! ³ ~ À b À! b À! b À! b! . What is ,´? c ? µ?
A) c À
B) À
C) À D) À b À
E) À b À! b Àc! b À! c!
26. Let ? be a continuous random variable with density function
²%³ ~ F
cO%O for c%
Á otherwise
.
Determine the density function of @ ~ ? where nonzero.
A) & c B) l& c &
C) l&
D) c l&
l
E)
l&
27. Let ? and @ be discrete random variables with joint probability function
²%Á &³given by the following table:
%
&
À
À
À
À
What is the variance of @ c ? ?
A) À
B) À
C) À
D) À
E) À
28. Let ? and ? be two independent observations from a normal distribution with
mean and variance 1. If ,´O? c ? O µ ~ , then ~
A) l
B)
l
C)
l
l
D)
E)
l
29. Let ? , @ and A be independent Poisson Random variables with ,´?µ ~ ,
,´@ µ ~ , and ,´Aµ ~ . What is 7 ´? b @ b A  µ ?
c°
A) c
B) c
C) D) c°
E) c°
30. Customers arrive randomly and independently at a service window, and the time
between arrivals has an exponential distribution with a mean of 12 minutes. Let ? equal
the number of arrivals per hour. What is 7 ´? ~ µ ?
c
A) [
c
B) [
c0
C) 2 [
c
D) 2 [
c
E) [
31. Let ? and @ be independent continuous random variables with common density
function
²!³ ~ F
for !
Á otherwise
What is 7 ´? ‚ @ µ ?
A) B) C) .
D) E) 32. Let ?Á @ and A have means Á and , respectively, and variances Á and ,
respectively. The covariance of ? and @ is 2, the covariance of ? and A is , and the
covariance of @ and A is 1. What are the mean and variance, respectively, of the random
variable ? b @ c A ?
A) 4 and 31
B) 4 and 65
C) 4 and 67
D) 14 and 13
E) 14 and 65
33. Let ? have a uniform distribution on the interval ²Á ³. What is the probability that
the sum of 2 independent observations of ? is greater than 5?
A) B) C) D) E) 34. Let A Á A Á A be independent random variables each with mean and variance
c , and let ? ~ A c A and @ ~ A b A . What is ?@ ?
A) c B) c C) c D) E) 35. The life (in days) of a certain machine has an exponential distribution with a mean of
1 day. The machine comes supplied with one spare. Find the density function (! measure
in days) of the combined life of the machine and its spare if the life of the spare has the
same distribution as the first machine, but is independent of the first machine.
A) !c!
B) c!
C) c!
D) ²! c ³c!
E) !c!
36. If ? ²%³ ~ %c% ° for % € , and @ ~ ? , find the density function for @ .
&
&
&
A) &c B) ² &³c² &³ °
C) &c D) &c& °
E) c PROBLEM SET 8 SOLUTIONS
?c?  !c? µ ~ 7 ´>  !c? µ , where > — 5 ²Á ³
@
c
@
!c
@
!c
@
-@ ²!³ ~ 7 ´@  !µ ~ 7 ´  µ ~ 7 ´=  µ , where = — 5 ²Á ³ À
!c? ‚ !c@ , which is equivalent to ?  @ À
-? ²!³ ‚ -@ ²!³ is equivalent to
1. -? ²!³ ~ 7 ´?  !µ ~ 7 ´
Note that the fact the ? and @ have a bivariate distribution with correlation coefficient
?@ is irrelevant - we are comparing probabilities of the marginal distributions of ? and
@ (however, we do use the fact that ? and @ have common variance ). Answer: B
? h ²? Á @ ³ ~ h b h b h b h ~ 2. ,´ ?
µ
~
@
%~ &~ @
Answer: D
3. It follows from the independence of ? and @ that
,´²? b ³ ²@ c ³ µ ~ ,´²? b ³ µ h ,´²@ c ³ µ À
,´²? b ³ µ ~ ,´? b ? b µ ~ ,´? µ b ,´?µ b Á and since
~ ,´? µ c ²,´?µ³ , we have ,´? µ ~ b ²,´?µ³ ~ , and then
?
?
,´²? b ³ µ ~ ,´? b ? b µ ~ b ²³ b ~ .
In a similar way, ,´@ µ ~ @ b ²,´@ µ³ ~ Á and
,´²@ c ³ µ ~ ,´@ c @ b µ ~ c ² c ³ b ~ ,
so that ,´²? b ³ ²@ c ³ µ ~ ,´²? b ³ µ h ,´²@ c ³ µ ~ h ~ .
Note that we could also find ,´²? b ³ µ in the following way:
²? b ³ ~ ? b ? b ~ ? c ? b b ? ~ ²? c ³ b ? , and then
b ? ~ (since
,´²? b ³ µ ~ ,´²? c ³ µ b ,´?µ ~ ?
= ´?µ ~ ,´²? c ? ³ µ Á and ? ~ ). ,´²@ c ³ µ can be found in a similar way.
Answer: E
4. The region of joint density is the triangular region above the line & ~ % and below
the horizontal line & ~ for  %  À The conditional density of & given ? ~ % is
²& O ? ~ %³ ~
²%Á&³ Á
? ²%³
where ? ²%³ is the marginal density function of %.
B
? ²%³ ~ cB ²%Á &³ & ~ % & ~ ² c %³ Á so that ²& O ? ~ %³ ~ ²c%³
~ c%
and the region of density for the conditional distribution of @ given ? ~ % is %  &  À
It is true in general that if a joint distribution is uniform (has constant density in a region)
then any conditional (though not necessarily marginal) distribution will be uniform on it
restricted region of probability - the conditional distribution of @ given ? ~ % is uniform
on the interval %  &  , with constant density c%
Answer: A
.
5. -? ²³ ~ 7 ´?  µ ~ lim - ²Á &³ ~ - ²Á ³ ~ ~ ,
&¦B
so that 7 ´? € µ ~ c 7 ´?  µ ~ À
Answer: C
6. *#´?Á @ µ ~ ,´?@ µ c ,´?µ h ,´@ µ ~ ,´?@ µ c À À
,´?@ µ ~ %& h ²%Á &³
%~ &~
~ h h ²À³ b h h ²À³ b Ä b h h ²³ b h h ²³ ~ À c À
~ c À À
Answer: B
7. Let 5 denote the number that appears on the wheel, so that
7 ´5 ~ µ ~ 7 ´5 ~ µ ~ 7 ´5 ~ µ ~ . Then, conditioning over 5 ,
7 ´>  Àµ ~ 7 ´>  ÀO5 ~ µ h 7 ´5 ~ µ b µ7 ´>  ÀO5 ~ µ h 7 ´5 ~ µ
b 7 ´>  ÀO5 ~ µ h 7 ´5 ~ µ .
If 5 ~ then > ~ , so that 7 ´>  ÀO5 ~ µ ~ Á and
if 5 ~ then > ~ , so that 7 ´>  ÀO5 ~ µ ~ .
If 5 ~ then > — 5 ²Á ³ so that
7 ´>  ÀO5 ~ µ ~ 7 ´ >c  Àc
O5 ~ µ ~ 7 ´A  c Àµ ~ À
(A has a standard normal distribution - the probability is found from the table).
Then, 7 ´>  Àµ ~ h b h b ²À³ h ~ À .
Answer: C
8. This discrete distribution has the following 8 points and probabilities:
; ²Á ³ - ; ²Á ³ - ; ²Á ³ - ; ²Á ³ - ; ²Á ³ - ;
²Á ³ - ; ²Á ³ - . The event ? b @  occurs at the points
²Á ³ - ²Á ³ Á ²Á ³ Á ²Á ³ Á ²Á ³ and ²Á ³ . The total probability of this event occurring is
Answer: E
b b b b ~ .
c%
9. 7 ´? € µ ~ ~ %² c %³ %
~ ²% c % ³ % ~
Answer: D
% & %
.
10. The marginal distribution of ? is
B
? ²%³ ~ cB ²%Á &³ &
cO%O & ~ c O%O for c  %  .
,´?µ ~ c %² c O%O³ %
~ c %² b %³ % b %² c %³ % ~ ,
,´? µ ~ c % ² c O%O³ %
~ % ² b %³ % b % ² c %³ %
c
b ~ . = ´?µ ~ ,´? µ c ²,´?µ³ ~ .
~ Answer: B
11. ? ²%³ ~ ²% b &³ & ~ % b .
Answer: E
12. = ´@ µ ~ = ´,´@ O?µµ b ,´= ´@ O?µµ .
We are given ,´@ O? ~ %µ ~ % and = ´@ O? ~ %µ ~ % , so that
,´@ O?µ ~ ? and = ´@ O?µ ~ ? , and then
= ´@ µ ~ = ´?µ b ,´? µ . We are given ,´?µ ~ , and = ´?µ ~ ,
so that ~ = ´?µ ~ ,´? µ c ²,´?µ³ S ,´? µ ~ , and therefore,
= ´@ µ ~ b ~ .
Answer: E
B
13. ? ²³ ~ 7 ´? ~ µ ~ ²Á &³ ~ ²Á ³ b ²Á ³ b ²Á ³ ~ .
&~cB
²Á³
°
Then we have conditional probabilities 7 ´@ ~ O? ~ µ ~ 7 ´?~µ ~ ° ~ Á
and similarly, 7 ´@ ~ O? ~ µ ~ and 7 ´@ ~ O? ~ µ ~ .
Then, ,´@ O? ~ µ ~ h b h b h ~ À
Answer: C
14. *#´?Á @ µ ~ ,´?@ µ c ,´?µ h ,´@ µ .
The region of probability is the triangle above
the line & ~ % in the unit square  %  ,
&.
&
,´?@ µ ~ %& h % % & ~ .
S *#´?Á @ µ ~ c h ~ Alternatively, ,´?@ µ ~ % %& h % & % ~ .
Answer: A
15. Since the total probability must be 1, we have b ~ .
The marginal distributions of ? and @ have
7 ´? ~ µ ~ 7 ´? ~ µ ~ 7 ´@ ~ µ ~ 7 ´@ ~ µ ~ b ~ À Then, because
of independence, 7 ´? ~ Á @ ~ µ ~ 7 ´? ~ µ h 7 ´@ ~ µ ~ ~ b .
Solving the two equations in and ( b ~ and b ~ ) results in
~ , ~ .
Answer: B
16. The region of probability is shown in the shaded figure below
%
b ~ .
The probability is %° %& & % b %° %& & % ~ &
Alternatively, the probability is & %& % & ~ ~ .
Answer: D
17. The marginal density of ? is ²%³ ~ ²%À&³ & ~ ²%O&³ h ²&³ & .
But ²%O&³ ~ & for  %  l& , or equivalently, for %  &  .
l
&~
Thus, ²%³ ~ % & h & ~ &° e ~ ² c %³ .
l
Answer: A
&~%
18. = ´@ O? ~ %µ ~ ,´@ O? ~ %µ c ²,´@ O? ~ %µ³ .
²%Á&³
@ O? ²&O? ~ %³ ~
, where ? ²%³ ~ & ~ ² c %³ À
? ²%³
%
so that ,´@ O? ~ %µ ~ & h & ~ b%
Thus, @ O? ²&O? ~ %³ ~ c%
% c%
b%b%
and ,´@ O? ~ %µ ~ % & h c% & ~
, and then
²c%³
b%b%
b%
= ´?µ ~
c ´ µ ~ .
Alternatively, note that given any joint uniform distribution, any related conditional
distribution is also uniform. Given ? ~ %, @ has a uniform distribution on ²%Á ³ and
²c%³
thus has a variance of .
Answer: B
19. The moment generating function of ? and ? is
(! c)(! c)
4 ²! Á ! ³ ~ ,´! ? b! ? µ ~ ! % b! % % % ~
. Answer: B
!!
2. The moment generating function for ? is 4? ²! ³ ~ 4 ²! Á ³ ~ b ! À
Then, ,´?µ ~ 4?Z ²³ ~ , and ,´? µ ~ 4?ZZ ²³ ~ , so that
= ´?µ ~ c ² ³ ~ .
Answer: D
21. The variance of a uniform random variable on the interval ´Á µ is
= ´? b ? c ? µ
~ = ´? µ b = ´? µ b = ´? µ
b h *#´? Á ? µ c *#´? Á ? µ c h *#´? Á ? µ
b b b c c ~ À
~ Answer: C
²c³ À
22. ? Á ? and ? are independent binomial random variables, all with the same value
of , and therefore, : ~ ? b ? b ? has a binomial distribution with parameters and ~ b b À The probability function of : is
b b b b c
7 ´: ~ µ ~ 8 9 ² c ³c ~ 8 9 ² c ³ À Answer: A
(for ? ‚ ) is a decreasing function, and
23. The transformation @ ~ "²?³ ~ ?c
therefore is invertible - ? ~ "c ²@ ³ ~ #²@ ³ ~ @ b . Then using the standard
method for finding the density of a transformed random variable, we have the density
#²&³ ~ ² b ³c h c
function of @ is ²&³ ~ ²#²&³³ h e &
e
e
&
& e ~ ²&b³ À
 &µ ~ 7 ´? ‚ b µ
Alternatively, -@ ²&³ ~ 7 ´@  &µ ~ 7 ´ ?c
&
B
~ ²&b³°& %c % ~ &b S @ ²&³ ~ -@Z ²&³ ~
&
²&b³ À
Answer: B
24. 4?b@ ²!³ ~ ,´!²?b@ ³ µ ~ ,´!? h !@ µ ~ ,´!? µ h ,´!@ µ ~ 4? ²!³ h 4@ ²!³
~ %²! b !³ h %´²!³ b !µ ~ %²! b !³ . Note that the equality
,´!? h !@ µ ~ ,´!? µ h ,´!@ µ follows from the independence of ? and @ .
Answer: E
25. ,´? µ ~ C!C 4 ²! Á ! ³e
C
and ,´? µ ~ C! 4 ²! Á ! ³e
! ~! ~
! ~! ~
C 4 ²! Á ! ³ ~ À! b À! b! S ,´? µ ~ À ,
C!
C
!
! b! S ,´? µ ~ À
,
C! 4 ²! Á ! ³ ~ À b À
S ,´? c ? µ ~ ,´? µ c ,´? µ ~ À .
Answer: B
26. For  &  , -@ ²&³ ~ 7 ´@  &µ ~ 7 ´?  &µ ~ 7 ´O?O  l & µ
l&
l&
~ cl& ² c O%O³ % ~ cl& ² b %³ % b ² c %³ % ~ l& c & À
Then, @ ²&³ ~ -@Z ²&³ ~ & c for  &  .
l
Note that in this case the transformation "²%³ ~ % is not one-to-one on the region of
#²&³
probability of ? ( c  %  ), we cannot use the @ ²&³ ~ ? ²#²&³³ h e &
e
approach.
Answer: A
27. The distribution of > ~ @ c ? is discrete with possible values Á c Á c Á and .
The probabilities are > ² c ³ ~ À (this occurs only if @ ~ and ? ~ c ),
> ² c ³ ~ À (@ ~ Á ? ~ ) , > ²³ ~ À (@ ~ Á ? ~ ) , and
> ²³ ~ À (@ ~ Á ? ~ ). Then ,´> µ ~ c À
, and ,´> µ ~ À , so that
= ´> µ ~ À c ² c À
³ ~ À .
Answer: C
28. > ~ ? c ? has a normal distribution with a mean of c ~ , and a variance
B
of b ~ . Then, ,´O? c ? O µ ~ ,´O> O µ ~ cB O$O h > ²$³ $
B
B
~ cB ² c $³> ²$³ $ b $ > ²$³ $ ~ $ > ²$³ $ .
But > ²$³ ~
h c$ ° (from the p.d.f. for 5 ²Á ³ ), so that
lhl
$~B
h c$ ° $ ~ c h c$ ° e
~ .
h $~
Thus, h ~ S ~ .
Answer: E
B $ > ²$³ $ ~ B $ h
l
l l
l
l
l
29. As the sum of independent Poisson random variables, > ~ ? b @ b A has a
Poisson distribution with parameter b b ~ , so that
c
7 ´>  µ ~ 7 ´> ~ µ b 7 ´> ~ µ ~ c b [h ~ c .
Answer: B
30. When the time between successive arrivals has an exponential distribution with mean
(units of time), then the number of arrivals per unit time has a Poisson distribution with
parameter (mean) . The time between successive arrivals has an exponential
distribution with mean hours (12 minutes). Thus, the number of arrivals per hour has a
c Poisson distribution with parameter , so that 7 ´? ~ µ ~ [ À Answer: E
31. Since both ? and @ are between 0 and 1, the event ? € @ is equivalent to
? € @ ° . Since ? and @ are independent, their joint density is
²%Á &³ ~ ? ²%³ h @ ²&³ ~ . Then,
7 ´? € @ µ ~ &° % & ~ ² c &° ³ & ~ À
Answer: C
32. ,´? b @ c Aµ ~ ,´?µ b ,´@ µ c ,´Aµ ~ ²³ b ²³ c ²³ ~ .
= ´? b @ c Aµ ~ = ´?µ b = ´@ µ b = ´Aµ
b ²
*#´?Á @ µ c *#´?Á Aµ c *#´@ Á Aµ³ ~ . Answer: C
33. The probability 7 ´? b ? € µ
is the integral of the joint density of ?
and ? over the shaded region at the right.
This region is  %  and
c %  %  . The probability is
h % % ~ À Answer: B
c% 34. ?@ ~
*#´?Á@ µ
? @ ~
*#´A cA ÁA bA µ
l= ´A cA µh= ´A bA µ .
*#´A c A Á A b A µ
~ *#´A Á A µ b *#´A Á A µ c *#´A Á A µ c *#´A Á A µ
~ ²³ b ²³ c ²³ c = ´A µ ~ c (since *#´A Á A µ ~ = ´A µ and
independent random variables have covariance of ).
= ´A c A µ ~ = ´A µ b = ´A µ c ²*#´A Á A µ³ ~ Á
= ´A b A µ ~ = ´A µ b = ´A µ b ²*#´A Á A µ³ ~ .
Thus, the correlation is ?@ ~ c ~ c . Answer: C
l²³h²³
35. ; ~ ; b ; , where ; is the random lifetime of machine (n days). Since ; and ;
are independent, the joint density of ; and ; is ²! Á ! ³ ~ c! c! .
Applying the convolution method for the sum of random variables results in
!
!
; ²!³ ~ ² Á ! c ³ ~ c c²!c ³ ~ !c! . Answer: A
36. The function & ~ "²%³ ~ % is strictly increasing (and thus, one-to-one) for all
% € , with the inverse function being % ~ & ~ #²&³ . Then
&
& @ ²&³ ~ ? ²#²&³³ h O#Z ²&³O ~ ? ²& ³ h & ~ & h c² ³ ° h & ~ &c .
Alternatively, -@ ²&³ ~ 7 ´@  &µ ~ 7 ´ ?  &µ ~ 7 ´?  & µ , and
% -? ²%³ ~ 7 ´?  %µ ~ !c! ° ! ~ c c% °
& & S -@ ²&³ ~ 7 ´?  & µ ~ !c! ° ! ~ c c² ³ °
S @ ²&³ ~ -@Z ²&³ ~ &c .
&
Answer: A