PROBABILITY NOTES - PR4
JOINT, MARGINAL, AND CONDITIONAL DISTRIBUTIONS
Joint distribution of random variables ? and @ : A joint distribution of two random
variables has a probability function or probability density function ²%Á &³ that is a function of
two variables (sometimes denoted ?Á@ ²%Á &³).
If ? and @ are discrete random variables, then ²%Á &³ must satisfy
(i)  ²%Á &³  and (ii) ²%Á &³ ~ .
% &
If ? and @ are continuous random variables, then ²%Á &³ must satisfy
B B
(i) ²%Á &³ ‚ and (ii) cB cB ²%Á &³ & % ~ .
It is possible to have a joint distribution in which one variable is discrete and one is continuous,
or either has a mixed distribution. The joint distribution of two random variables can be
extended to a joint distribution of any number of random variables.
If ( is a subset of two-dimensional space, then 7 ´²?Á @ ³  (µ is the double summation
(discrete case) or double integral (continuous case) of ²%Á &³ over the region (.
Cumulative distribution function of a joint distribution: If random variables ? and @ have a
joint distribution, then the cumulative distribution function is
- ²%Á &³ ~ 7 ´²?  %³ q ²@  &³µ .
% &
In the continuous case, - ²%Á &³ ~ cB cB ² Á !³ ! ,
%
and in the discrete case, - ²%Á &³ ~ &
² Á !³ .
~cB !~cB
In the continuous case, C%C C& - ²%Á &³ ~ ²%Á &³ .
Expectation of a function of jointly distributed random variables: If ²%Á &³ is a function of
two variables, and ? and @ are jointly distributed random variables, then
the expected value of ²?Á @ ³ is defined to be
,´²?Á @ ³µ ~ ²%Á &³ h ²%Á &³ in the discrete case, and
% &
B B
,´²?Á @ ³µ ~ cB cB ²%Á &³ h ²%Á &³ & % in the continuous case.
1
Marginal distribution of ? found from a joint distribution of ? and @ :
If ? and @ have a joint distribution with joint density or probability function ²%Á &³, then the
marginal distribution of ? has a probability function or density function denoted ? ²%³, which
B
is equal to ? ²%³ ~ ²%Á &³ in the discrete case, and is equal to ? ²%³ ~ ²%Á &³ &
cB
&
in the continuous case. The density function for the marginal distribution of @ is found in a
B
similar way ; @ ²&³ is equal to either @ ²&³ ~ ²%Á &³ or @ ²&³ ~ cB ²%Á &³ % .
%
If the cumulative distribution function of the joint distribution of ? and @ is - ²%Á &³, then
-? ²%³ ~ lim - ²%Á &³ and -@ ²&³ ~ lim - ²%Á &³ .
&¦B
%¦B
This can be extended to define the marginal distribution of any one (or subcollection) variable in
a multivariate distribution.
Independence of random variables ? and @ : Random variables ? and @ with cumulative
distribution functions -? ²%³ and -@ ²&³ are said to be independent (or stochastically
independent) if the cumulative distribution function of the joint distribution - ²%Á &³ can be
factored in the form - ²%Á &³ ~ -? ²%³ h -@ ²&³ for all ²%Á &³. This
definition can be extended to a multivariate distribution of more than 2 variables.
If ? and @ are independent, then ²%Á &³ ~ ? ²%³ h @ ²&³ , (but the reverse implication
is not always true, i.e. if the joint distribution probability or density function can be factored in
the form ²%Á &³ ~ ? ²%³ h @ ²&³ then ? and @ are usually, but not always, independent).
Conditional distribution of @ given ? ~ %: Suppose that the random variables ? and @ have
joint density/probability function ²%Á &³, and the density/probability function of the marginal
distribution of ? is ? ²%³. The density/probability function of the conditional distribution of @
²%Á&³
given ? ~ % is @ O? ²&O? ~ %³ ~ ²%³ , if ? ²%³ £ .
?
B
The conditional expectation of @ given ? ~ % is ,´@ O? ~ %µ ~ cB & h @ O? ²&O? ~ %³ & in
the continuous case, and ,´@ O? ~ %µ ~ & h @ O? ²&O? ~ %³ in the discrete case.
%
The conditional density/probability is also written as @ O? ²&O%³ , or ²&O%³ .
If ? and @ are independent random variables, then @ O? ²&O? ~ %³ ~ @ ²&³ and
?O@ ²%O@ ~ &³ ~ ? ²%³ .
Covariance between random variables ? and @ : If random variables ? and @ are jointly
distributed with joint density/probability function ²%Á &³, then the covariance between ? and @
is *#´?Á @ µ ~ ,´²? c ,´?µ³²@ c ,´@ µ³µ ~ ,´²? c ? ³²@ c @ ³µ .
Note that *#´?Á ?µ ~ = ´?µ .
2
Coefficient of correlation between random variables ? and @ :
The coefficient of correlation between random variables ? and @ is
²?Á @ ³ ~ ?Á@ ~
*#´?Á@ µ
? @
, where ? and @ are the standard deviations of ? and
@ respectively.
Moment generating function of a joint distribution: Given jointly distributed random
variables ? and @ , the moment generating function of the joint distribution is
4?Á@ ²! Á ! ³ ~ ,´! ?b! @ µ . This definition can be extended to the joint distribution of
any number of random variables.
Multinomial distribution with parameters Á Á Á À À ÀÁ (where is a positive
integer and   for all ~ Á Á ÀÀÀÁ and b b Ä ~ ³:
Suppose that an experiment has possible outcomes, with probabilities Á Á ÀÀÀÁ respectively. If the experiment is performed successive times (independently), let ? denote
the number of experiments that resulted in outcome , so that
? b ? b Ä b ? ~ . The multivariate probability function is
²% Á % Á ÀÀÀÁ % ³ ~ % [h%[[Ä% [ h % h % Ä% .
,´? µ ~ Á = ´? µ ~ ² c ³ Á *#´? ? µ ~ c .
For example, the toss of a fair die results in one of ~ 6 outcomes, with probabilities
~ for ~ Á Á Á Á Á . If the die is tossed times, then with
? ~ # of tosses resulting in face "" turning up, the multivariate distribution of ? Á ? Á ÀÀÀÁ ?
is
a multinomial distribution.
Some results and formulas related to this section are:
(i) ,´ ²?Á @ ³ b ²?Á @ ³µ ~ ,´ ²?Á @ ³µ b ,´ ²?Á @ ³µ , and in particular,
,´? b @ µ ~ ,´?µ b ,´@ µ and ,´? µ ~ ,´? µ
(ii) lim - ²%Á &³ ~ lim - ²%Á &³ ~ %¦cB
&¦cB
(iii) 7 ´²%  ?  % ³ q ²&  @  & ³µ
~ - ²% Á & ³ c - ²% Á & ³ c - ²% Á & ³ b - ²% Á & ³
(iv) 7 ´²?  %³ q ²@  &³µ ~ -? ²%³ b -@ ²&³ c - ²%Á &³  3
(v) If ? and @ are independent, then for any functions and ,
,´²?³ h ²@ ³µ ~ ,´²?³µ h ,´²@ ³µ , and in particular, ,´? h @ µ ~ ,´?µ h ,´@ µ.
(vi) The density/probability function of jointly distributed variables ? and @ can be
written in the form ²%Á &³ ~ @ O? ²&O? ~ %³ h ? ²%³ ~ ?O@ ²%O@ ~ &³ h @ ²&³
(vii) *#´?Á @ µ ~ ,´? h @ µ c ? h @ ~ ,´?@ µ c ,´?µ h ,´@ µ .
*#´?Á @ µ ~ *#´@ Á ?µ . If ? and @ are independent, then ,´? h @ µ ~ ,´?µ h ,´@ µ
and *#´?Á @ µ ~ . For constants Á Á Á Á Á and random variables ?Á @ Á A
and > , *#´? b @ b Á A b > b µ
~ *#´?Á Aµ b *#´?Á > µ b *#´@ Á Aµ b *#´@ Á > µ
(viii) For any jointly distributed random variables ? and @ , c  ?@  (ix) = ´? b @ µ ~ ,´²? b @ ³ µ c ²,´? b @ µ³
~ ,´? b ?@ b @ µ c ²,´?µ b ,´@ µ³
~ ,´? µ b ,´?@ µ b ,´@ µ c ²,´?µ³ c ,´?µ,´@ µ c ²,´@ µ³
~ = ´?µ b = ´@ µ b h *#´?Á @ µ
If ? and @ are independent, then = ´? b @ µ ~ = ´?µ b = ´@ µ .
For any ?Á @ , = ´? b @ b µ ~ = ´?µ b = ´@ µ b h *#´?Á @ µ
(x) 4?Á@ ²! Á ³ ~ ,´! ? µ ~ 4? ²! ³ and 4?Á@ ²Á ! ³ ~ ,´! @ µ ~ 4@ ²! ³
(xi) C!C 4?Á@ ²! Á ! ³e
~ ,´?µ , C!C 4?Á@ ²! Á ! ³e
~ ,´@ µ
! ~! ~
! ~! ~
b
C
C ! C ! 4?Á@ ²! Á ! ³e! ~! ~ ~ ,´? h @ µ
(xii) If 4 ²! Á ! ³ ~ 4 ²! Á ³ h 4 ²Á ! ³ for ! and ! in a region about ²Á ³,
then ? and @ are independent.
(xiii) If @ ~ ? b then 4@ ²!³ ~ ! 4? ²!³ .
(xiv) If ? and @ are jointly distributed, then for any &, ,´?O@ ~ &µ depends on &, say
,´?O@ ~ &µ ~ ²&³ . It can then be shown that ,´²@ ³µ ~ ,´?µ ; this is more
usually written in the form ,´ ,´?O@ µ µ ~ ,´?µ .
It can also be shown that = ´?µ ~ ,´ = ´?O@ µ µ b = ´ ,´?O@ µ µ .
4
(xv) A random variable ? can be defined as a combination of two (or more) random
variables ? and ? , defined in terms of whether or not a particular event ( occurs.
?~F
? if event ( occurs (probability )
? if event ( does not occur (probability c )
Then, @ can be the indicator random variable 0( ~ F
if ( occurs (prob. )
if ( doesn't occur (prob. c)
Probabilities and expectations involving ? can be found by "conditioning" over @ :
7 ´?  µ ~ 7 ´?  O( occursµ h 7 ´( occursµ b 7 ´?  O(Z occursµ h 7 ´(Z occursµ
~ 7 ´?  µ h b 7 ´?  µ h ² c ³ ,
,´? µ ~ ,´? µ h b ,´? µ h ² c ³ Á 4? ²!³ ~ 4? ²!³ h b 4? ²!³ h ² c ³
This is really an illustration of a mixture of the distributions of ? and ? , with ~ and ~ c .
As an example, suppose there are two urns containing balls - Urn I contains 5 red and 5
blue balls and Urn II contains 8 red and 2 blue balls. A die is tossed, and if the number
turning up is even then 2 balls are picked from Urn I, and if the number turning up is
odd then 3 balls are picked from Urn II. ? is the number of red balls chosen. Event (
would be ( ~ die toss is even. Random variable ? would be the number of red
balls chosen from Urn I and ? would be the number of red balls chosen from Urn II, and
since each urn is equally likely to be chosen, ~ ~ À
(xvi) If ? and @ have a joint distribution which is uniform on the two dimensional
region 9 (usually 9 will be a triangle, rectangle or circle in the ²%Á &³ plane), then the
conditional distribution of @ given ? ~ % has a uniform distribution on the line
segment (or segments) defined by the intersection of the region 9 with the line ? ~ %.
The marginal distribution of @ might or might not be uniform.
Example 116: If ²%Á &³ ~ 2²% b & ³ is the density function for the joint distribution of the
continuous random variables ? and @ defined over the unit square bounded by the points ²Á ³Á
²Á ³ Á ²Á ³ and ²Á ³, find 2 .
Solution: The (double) integral of the density function over the region of density must be 1, so
that ~ 2²% b & ³ & % ~ 2 h S 2 ~ .
U
Example 117: The cumulative distribution function for the joint distribution of the continuous
random variables ? and @ is - ²%Á &³ ~ ²À³²% & b % & ³ , for  %  and  &  .
Find ² Á ³ .
Solution: ²%Á &³ ~ C%C C& - ²%Á &³ ~ ²À³²% b %&³ S ² Á ³ ~ À
5
U
Example 118: ? and @ are discrete random variables which are jointly distributed
with the following probability function ²%Á &³:
?
c
@
c
Find ,´? h @ µ.
Solution: ,´?@ µ ~ %& h ²%Á &³ ~ ² c ³²³² ³ b ² c ³²³² ³ b ² c ³² c ³² ³
% &
b ²³²³² ³ b ²³²³²³ b ²³² c ³² ³
b ²³²³² ³ b ²³²³² ³ b ²³² c ³² ³ ~ .
U
Example 119: Continuous random variables ? and @ have a joint distribution with density
function ²%Á &³ ~
²c%c&³
in the region bounded by & ~ Á % ~ and & ~ c %. Find the density function for the marginal distribution of ? for  %  .
Solution: The region of joint density is illustrated
in the graph at the right. Note that ? must be in the
interval ²Á ³ and @ must be in the interval ²Á ³.
B
Since ? ²%³ ~ cB ²%Á &³ & , we note that
given a value of % in ²Á ³, the possible values of &
(with non-zero density for ²%Á &³) must satisfy
 &  c %, so that
c% ²%Á &³ &
? ²%³ ~ c% ²c%c&³ & ~ ² c %³ . U
~ Example 120: Suppose that ? and @ are independent continuous random variables with the
following density functions - ? ²%³ ~ for  %  and
@ ²&³ ~ & for  &  . Find 7 ´@  ?µ.
Solution: Since ? and @ are independent, the
density function of the joint distribution of ? and
@ is ²%Á &³ ~ ? ²%³ h @ ²&³ ~ & , and is
defined on the unit square. The graph at the
right illustrates the region for the probability
%
in question. 7 ´@  ?µ ~ & & % ~ . U
6
Example 121: Continuous random variables ? and @ have a joint distribution with density
function ²%Á &³ ~ % b for  %  and  &  .
Find 7 ´? € O@ € µ.
%&
7 ´²?€ ³q²@ € ³µ
.
7 ´@ € µ
%&
7 ´²? € ³ q ²@ € ³µ ~ ° ° ´% b µ & % ~ .
%&
7 ´@ € µ ~ ° @ ²&³ & ~ ° ´
²%Á &³ %µ & ~ ° ´% b µ % & ~ S
°
7 ´? € O@ € µ ~ °
~ U
.
Solution: 7 ´? € O@ € µ ~
Example 122: Continuous random variables ? and @ have a joint distribution with density
function ²%Á &³ ~ ² &³c% for  %  B and  &  .
Find 7 ´? € O@ ~ µ .
7 ´²?€³q²@ ~ ³µ
Solution: 7 ´? € O@ ~ µ ~
.
@ ² ³
l
B
B
7 ´²? € ³ q ²@ ~ ³µ ~ ²%Á ³ % ~ ² ³c% % ~ h c .
B
B
@ ² ³ ~ ²%Á ³ % ~ ² ³c% % ~ S 7 ´? € O@ ~ µ ~ c .
U
l
Example 123: ? is a continuous random variable with density function ? ²%³ ~ % b for
 %  . ? is also jointly distributed with the continuous random variable @ , and the
conditional density function of @ given ? ~ % is
%b&
@ O? ²&O? ~ %³ ~
for  %  and  &  . Find @ ²&³ for  &  .
%b Solution: ²%Á &³ ~ ²&O%³ h ? ²%³ ~
Then, @ ²&³ ~ ²%Á &³ % ~ & b À
%b&
h ²% b ³ ~ % b & .
%b U
Example 124: Find *#´?Á @ µ for the jointly distributed discrete random variables in
Example 118 above.
Solution: *#´?Á @ µ ~ ,´?@ µ c ,´?µ h ,´@ µ . In Example 118 it was found that
,´?@ µ ~ . The marginal probability function for ? is 7 ´? ~ µ ~ b b ~ ,
7 ´? ~ µ ~ and 7 ´? ~ c µ ~ , and the mean of ? is
,´?µ ~ ²³² ³ b ²³² ³ b ² c ³² ³ ~ .
In a similar way, the probability function of @ is found to be 7 ´@ ~ µ ~ Á 7 ´@ ~ µ ~ Á
and 7 ´@ ~ c µ ~ Á with a mean of ,´@ µ ~ c .
Then, *#´?Á @ µ ~ c ² ³² c ³ ~ U
.
7
Example 125: The coefficient of correlation between random variables ? and @ is ,
~ Á ~ . The random variable A is defined to be A ~ ? c @ , and
and ?
@
it is found that A ~ . Find .
Solution: A ~ = ´Aµ ~ = ´?µ b = ´@ µ c h ²³²³ *#´?Á @ µ .
Since *#´?Á @ µ ~ ´?Á @ µ h ? h @ ~ h l h l ~ Á it follows that
~ A ~ b ²³ c ² ³ ~ S ~ .
U
Example 126: Suppose that ? and @ are random variables whose joint distribution has moment
generating function 4 ²! Á ! ³ ~ ² ! b ! b ³ , for all real ! and ! .
Find the covariance between ? and @ .
Solution: *#´?Á @ µ ~ ,´?@ µ c ,´?µ h ,´@ µ .
,´?@ µ ~ C !C C ! 4?Á@ ²! Á ! ³e
! ~! ~
~ ²³²³² ! b ! b ³ ² ! ³² ! ³e
~ ,
! ~! ~ ~ ²³² ! b ! b ³ ² ! ³e
~
! ~! ~
! ~! ~
,´@ µ ~ C!C 4?Á@ ²! Á ! ³e
~ ²³² ! b ! b ³ ² ! ³e
~
,´?µ ~ C!C 4?Á@ ²! Á ! ³e
! ~! ~
! ~! ~
S *#´?Á @ µ ~ c ² ³² ³ ~ c . U
Á
Á
Example 127: Suppose that ? has a continuous distribution with p.d.f. ? ²%³ ~ %
on the interval ²Á ³ , and ? ²%³ ~ elsewhere. Suppose that @ is a continuous random
variable such that the conditional distribution of @ given ? ~ % is uniform on the interval ²Á %³
. Find the mean and variance of @ .
Solution: This problem can be approached in two ways.
(i) The first approach is to determine the unconditional (marginal) distribution of @ . We are
given ? ²%³ ~ % for  %  , and @ O? ²&O? ~ %³ ~ % for  &  % .
Then, ²%Á &³ ~ ²&O%³ h ? ²%³ ~ % h % ~ for  %  and  &  % .
The unconditional (marginal) distribution of @ has p.d.f.
B
@ ²&³ ~ cB ²%Á &³ % ~ & % ~ ² c &³ for  &  (and @ ²&³ is elsewhere). Then ,´@ µ ~ & h ² c &³ & ~ , ,´@ µ ~ & h ² c &³ & ~ ,
and = ´@ µ ~ ,´@ µ c ²,´@ µ³ ~ c ² ³ ~ .
8
(ii) The second approach is to use the relationships ,´@ µ ~ ,´ ,´@ O?µ µ and
= ´@ µ ~ ,´ = ´@ O?µ µ b = ´ ,´@ O?µ µ .
From the conditional density ²&O? ~ %³ ~ % for  &  % , we have
%
,´@ O? ~ %µ ~ & h & ~ % , so that ,´@ O?µ ~ ? , and, since ? ²%³ ~ %,
%
%
,´ ,´@ O?µ µ ~ ,´ ?
µ ~ h %% ~ ~ ,´@ µ .
In a similar way, = ´@ O? ~ %µ ~ ,´@ O? ~ %µ c ²,´@ O? ~ %µ³ ,
%
where ,´@ O? ~ %µ ~ & h & ~ % , so that ,´@ O?µ ~ ? , and since
%
?
?
? ?
,´@ O?µ ~ , we have = ´@ O?µ ~ c ² ³ ~ .
%
Then ,´ = ´@ O?µ µ ~ ,´ ?
µ ~ h % % ~ , and
= ´,´@ O?µµ ~ = ´ ?
µ ~ = ´?µ ~ h ´,´? µ c ²,´?µ³ µ ~ h ´ c ² ³ µ ~ so that ,´ = ´@ O?µ µ b = ´ ,´@ O?µ µ ~ b ~ ~ = ´@ µ . U
9
FUNCTIONS AND TRANSFORMATIONS OF RANDOM VARIABLES
Distribution of a function of a continuous random variable ? : Suppose that ? is a
continuous random variable with p.d.f. ? ²%³ and c.d.f. -? ²%³, and suppose that "²%³ is a oneto-one function (usually " is either strictly increasing, such as "²%³ ~ % Á % Á l% or %, or "
is strictly decreasing, such as "²%³ ~ c% ). As a one-to-one function, " has an inverse function
#, so that #²"²%³³ ~ % . Then the random variable @ ~ "²?³ (@ is referred to as a
transformation of ?) has p.d.f. @ ²&³ found as follows: @ ²&³ ~ ? ²#²&³³ h O#Z ²&³O . If " is a
strictly increasing function, then
-@ ²&³ ~ 7 ´@  &µ ~ 7 ´"²?³  %µ ~ 7 ´?  #²&³µ ~ -? ²#²&³³ .
Distribution of a function of a discrete random variable ? : Suppose that ? is a discrete
random variable with probability function ²%³. If "²%³ is a function of %,
and @ is a random variable defined by the equation @ ~ "²?³ , then @ is a discrete random
variable with probability function ²&³ ~ ²%³ - given a value of &, find all values of % for
&~"²%³
which & ~ "²%³ (say "²% ³ ~ "²% ³ ~ Ä ~ "²%! ³ ~ &³, and then ²&³ is the sum of those
²% ³ probabilities.
If ? and @ are independent random variables, and " and # are functions, then the random
variables "²?³ and #²@ ³ are independent.
The distribution of a sum of random variables:
(i) If ? and ? are random variables, and @ ~ ? b ? , then
,´@ µ ~ ,´? µ b ,´? µ and = ´@ µ ~ = ´? µ b = ´? µ b *#´? Á ? µ
(ii) If ? and ? are discrete non-negative integer valued random variables with joint
probability function ²% Á % ³ , then for an integer ‚ ,
7 ´? b ? ~ µ ~ ²% Á c % ³ (this considers all combinations of ? and ?
% ~
whose sum is ).
If ? and ? are independent with probability functions ²% ³ and ²% ³, respectively,
then 7 ´? b ? ~ µ ~ ²% ³ h ² c % ³ (this is the convolution method of
% ~
finding the distribution of the sum of independent discrete random variables).
10
(iii) If ? and ? are continuous random variables with joint density function ²% Á % ³
B
then the density function of @ ~ ? b ? is @ ²&³ ~ ²% Á & c % ³ % .
cB
If ? and ? are independent continuous random variables with density functions
²% ³ and ²% ³Á then the density function of @ ~ ? b ? is
B
@ ²&³ ~ cB ²% ³ h ²& c % ³ %
(iv) If ? Á ? Á ÀÀÀÁ ? are random variables, and the random variable @ is defined to be
@ ~ ? , then ,´@ µ ~ ,´? µ and
~
~
~
~~b
= ´@ µ ~ = ´? µ b *#´? Á ? µ .
If ? Á ? Á ÀÀÀÁ ? are mutually independent random variables, then
~
~
= ´@ µ ~ = ´? µ and 4@ ²!³ ~ 4? ²!³ ~ 4? ²!³ h 4? ²!³Ä4? ²!³
(v) If ? Á ? Á ÀÀÀÁ ? and @ Á @ Á ÀÀÀÁ @ are random variables and
Á Á ÀÀÀÁ Á Á Á Á ÀÀÀÁ and are constants, then
~
~
~ ~
*#´ ? b Á @ b µ ~ *#´? Á @ µ
(vi) The Central Limit Theorem: Suppose that ? is a random variable with mean and standard deviation and suppose that ? Á ? Á ÀÀÀÁ ? are independent random
variables with the same distribution as ? . Let @ ~ ? b ? b Ä b ? . Then
,´@ µ ~ and = ´@ µ ~ , and as increases, the distribution of @ approaches
a normal distribution 5 ²Á ³. This is a justification for using the normal
distribution as an approximation to the distribution of a sum of random variables.
(vii) Sums of certain distributions: Suppose that ? Á ? Á ÀÀÀÁ ? are independent
random variables and @ ~ ?
distribution of ?
Bernoulli )²Á ³
binomial )² Á ³
~
distribution of @
binomial )²Á ³
binomial )² Á ³
geometric negative binomial Á negative binomial Á negative binomial Á Poisson Poisson 5 ² Á ³
5 ² Á ³
11
Example 128: The random variable ? has an exponential distribution with a mean of 1. The
random variable @ is defined to be @ ~ ? . Find @ ²&³, the p.d.f. of @ .
Solution: -@ ²&³ ~ 7 ´@  &µ ~ 7 ´ ?  &µ ~ 7 ´?  &° µ ~ c c
&°
&°
S @ ²&³ ~ -@Z ²&³ ~ &
² c c ³ ~ &° h c .
&°
Alternatively, since @ ~ ? (& ~ "²%³ ~ %, and is a strictly increasing function
with inverse % ~ #²&³ ~ &° ), and ? ~ @ ° , it follows that
&°
&°
@ ²&³ ~ ? ²&° ³ h e &
e ~ &° h c .
U
Example 129: Suppose that ? and @ are independent discrete integer-valued random variables
with ? uniformly distributed on the integers to , and @ having the following probability
function - @ ²³ ~ À , @ ²³ ~ À , @ ²³ ~ À . Let A ~ ? b @ .
Find 7 ´A ~ µ .
Solution: Using the fact that ? ²%³ ~ À for % ~ Á Á Á Á , and the convolution method for
independent discrete random variables, we have
A ²³ ~ ? ²³ h @ ² c ³
~
~ ²³²³ b ²À³²³ b ²À³²À³ b ²À³²³ b ²À³²À³ b ²À³²À³ ~ À .
U
Example 130: ? and ? are independent exponential random variables each with a mean of 1.
Find 7 ´? b ?  µ.
Solution: Using the convolution method, the density function of @ ~ ? b ? is
&
&
@ ²&³ ~ ? ²!³ h ? ²& c !³ ! ~ c! h c²&c!³ ! ~ &c& , so that
&~
~ c c
&~
7 ´? b ?  µ ~ 7 ´@  µ ~ &c& & ~ ´ c &c& c c& µe
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(the last integral required integration by parts).
Example 131: Given independent random variables ? Á ? Á ÀÀÀÁ ? each having the same
variance of , and defining < ~ ? b ? b Ä b ?c and
= ~ ? b ? b Ä b ? , find the coefficient of correlation between < and = .
; < ~ ² b b b Ä b ³ ~ ² b ³ ~ = .
Since the ? 's are independent, if £ then *#´? Á ? µ ~ . Then, noting that
*#´> Á > µ ~ = ´> µ, we have
*#´< Á = µ ~ *#´? Á ? µ b *#´? Á ? µ b Ä b *#´?c Á ? µ
Solution: <= ~
*#´< Á= µ
< =
~ = ´? µ b = ´? µ b Ä b = ´?c µ ~ ² c ³ .
²c³
Then, <= ~ ²b³ ~ c
b À
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Example 132: Independent random variables ?Á @ and A are identically distributed. Let
> ~ ? b @ . The moment generating function of > is 4> ²!³ ~ ²À b À! ³
.
Find the moment generating function of = ~ ? b @ b A .
Solution: For independent random variables,
4?b@ ²!³ ~ 4? ²!³ h 4@ ²!³ ~ ²À b À! ³
. Since ? and @ have identical
distributions, they have the same moment generating function. Thus,
4? ²!³ ~ ²À b À! ³ , and then 4= ²!³ ~ 4? ²!³ h 4@ ²!³ h 4A ²!³ ~ ²À b À! ³ .
Alternatively, note that the moment generating function of the binomial )²Á ³ is
² c b ! ³ . Thus, ? b @ has a )²
Á À³ distribution, and each of ?Á @ and A has
a )²Á À³ distribution, so that the sum of these independent binomial distributions is
)²Á À³ , with m.g.f. ²À b À! ³ .
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Example 133: The birth weight of males is normally distributed with mean 6 pounds, 10
ounces, standard deviation 1 pound. For females, the mean weight is 7 pounds, 2 ounces with
standard deviation 12 ounces. Given two independent male/female births, find the probability
that the baby boy outweighs the baby girl.
Solution: Let random variables ? and @ denote the boy's weight and girl's weight, respectively.
Then, > ~ ? c @ has a normal distribution with mean
c ~ c lb. and variance ? b @ ~ b ~ À
Then, 7 ´? € @ µ ~ 7 ´? c @ € µ ~ 7 >
> c²c ³
c²c ³
€
? ~ 7 ´A € Àµ,
°
°
l
l
where A has standard normal distribution (> was standardized). Referring to the
standard normal table, this probability is À .
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Example 134: If the number of typographical errors per page type by a certain typist follows a
Poisson distribution with a mean of , find the probability that the total number of errors in 10
randomly selected pages is 10.
Solution: The 10 randomly selected pages have independent distributions of errors per page.
The sum of independent Poisson random variables with parameters
Á ÀÀÀÀÁ has a Poisson distribution with parameter . Thus, the total number
of errors in the 10 randomly selected pages has a Poisson distribution with parameter .
The probability of 10 errors in the 10 pages is
c
²
³
.
[
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