chapter 5 000
TRANSCRIPT
-
8/9/2019 Chapter 5 000
1/31
Chapter 5: Joint Probability Distributions and Random Samples
CHAPTER 5
Section 5.1
1.
a. P(X = 1, Y = 1 = p(1,1 = !"#
b. P(X ≤ 1 and Y ≤ 1 = p(#,# $ p(#,1 $ p(1,# $ p(1,1 = !%"
c. &t least one hose is in use at both islands! P(X ≠ # and Y ≠ # = p(1,1 $ p(1," $ p(",1
$ p("," = !'#
d. y summin) ro* probabilities, p+(+ = !1, !-%, !5# .or + = #, 1, ", and by summin)
/olumn probabilities, py(y = !"%, !-0, !-0 .or y = #, 1, "! P(X ≤ 1 = p+(# $ p+(1 = !5#
e. P(#,# = !1#, but p+(# ⋅ py(# = (!1(!"% = !#-0% ≠ !1#, so X and Y are not independent!
2.
a.
y
p(+,y # 1 " - %
# !-# !#5 !#"5 !#"5 !1# !5
+ 1 !10 !#- !#15 !#15 !# !-
" !1" !#" !#1 !#1 !#% !"
! !1 !#5 !#5 !"
b. P(X ≤ 1 and Y ≤ 1 = p(#,# $ p(#,1 $ p(1,# $ p(1,1 = !5= (!0(!' = P(X ≤ 1 ⋅ P(Y ≤ 1
c. P( X $ Y = # = P(X = # and Y = # = p(#,# = !-#
d. P(X $ Y ≤ 1 = p(#,# $ p(#,1 $ p(1,# = !5-
3.
a. p(1,1 = !15, the entry in the 1st ro* and 1st /olumn o. the oint probability table!
b. P( X1 = X" = p(#,# $ p(1,1 $ p("," $ p(-,- = !#0$!15$!1#$!#' = !%#
c. & = 2 (+1, +": +1 ≥ " $ +" 3 ∪ 2 (+1, +": +" ≥ " $ +1 3P(& = p(",# $ p(-,# $ p(%,# $ p(-,1 $ p(%,1 $ p(%," $ p(#," $ p(#,- $ p(1,- =!""
d. P( e+a/tly % = p(1,- $ p("," $ p(-,1 $ p(%,# = !1'P(at least % = P(e+a/tly % $ p(%,1 $ p(%," $ p(%,- $ p(-," $ p(-,- $ p(",-=!%
1'5
-
8/9/2019 Chapter 5 000
2/31
-
8/9/2019 Chapter 5 000
3/31
Chapter 5: Joint Probability Distributions and Random Samples
c. p(+,y = # unless y = #, 1, ;, +< + = #, 1, ", -, %! or any su/h pair,
p(+,y = P(Y = y 8 X = + ⋅ P(X = + = (%(!(! x p y
x x
y x y ⋅
−
py(% = p(y = % = p(+ = %, y = % = p(%,% = (!%⋅(!15 = !#14%
py(- = p(-,- $ p(%,- = 1#50!15(!%(!(!-
%"5(!(! -- = +
py(" = p("," $ p(-," $ p(%," = "5(!%(!(!"
--(!(! ""
+
"'0!15(!%(!(!"
%"" =
+
py(1 = p(1,1 $ p(",1 $ p(-,1 $ p(%,1 = -(!%(!(!1
""(!(!
+
-54#!15(!%(!(!1
%"5(!%(!(!
1
--" =
+
py(# = 1 > ?!-54#$!"'0$!1#50$!#14%@ = !"%0#
7.
a. p(1,1 = !#-#
b. P(X ≤ 1 and Y ≤ 1 = p(#,# $ p(#,1 $ p(1,# $ p(1,1 = !1"#
c. P(X = 1 = p(1,# $ p(1,1 $ p(1," = !1##< P(Y = 1 = p(#,1 $ ; $ p(5,1 = !-##
d. P(o6er.lo* = P(X $ -Y 5 = 1 > P(X $ -Y ≤ 5 = 1 > P?(X,Y=(#,# or ;or (5,# or
(#,1 or (1,1 or (",1@ = 1 A !"# = !-0#
e. Bhe mar)inal probabilities .or X (ro* sums .rom the oint probability table are p+(# = !#5, p+(1 = !1# , p+(" = !"5, p+(- = !-#, p+(% = !"#, p+(5 = !1#< those .or Y (/olumnsums are py(# = !5, py(1 = !-, py(" = !"! t is no* easily 6eri.ied that .or e6ery (+,y,
p(+,y = p+(+ ⋅ py(y, so X and Y are independent!
1''
-
8/9/2019 Chapter 5 000
4/31
Chapter 5: Joint Probability Distributions and Random Samples
8.
a. numerator = ( )( ) ( ) "%#,-#1"%551
1"
"
1#
-
0==
denominator = ''5,54-
-#=
< p(-," = #5#4!
''5,54-
"%#,-# =
b. p(+,y =
( )
+−
#
-#
1"1#0
y x y x
otherwise
y x
that suchegers
negativenonare y x
#
D D int
D D ,
≤+≤
−
9.
a.
∫ ∫ ∫ ∫ +==
∞
∞−
∞
∞−
-#
"#
-#
"#
"" (,(1 dxdy y x K dxdy y x f
∫ ∫ ∫ ∫ ∫ ∫ +=+=-#
"#
"-#
"#
"-#
"#
-#
"#
"-#
"#
-#
"#
"1#1# dy y K dx x K dxdy y K dydx x K
###,-0#
-
-
###,14"# =⇒
⋅= K K
b. P(X E " and Y E " = ∫ ∫ ∫ =+"
"#
""
"#
"
"#
"" 1"( dx x K dxdy y x K
-#"%!-#%,-0%"
"#
- == K Kx
c.
P( 8 X > Y 8 ≤ " = ∫∫ III region
dxdy y x f ,(
∫∫ ∫∫ −−
II I
dxdy y x f dxdy y x f ,(,(1
∫ ∫ ∫ ∫ −
+−−
-#
""
"
"#
"0
"#
-#
",(,(1
x
xdydx y x f dydx y x f
= (a.ter mu/h al)ebra !-54-
1'0
I
II
"+= x y "−= x y
"#
"#
-#
-#
III
-
8/9/2019 Chapter 5 000
5/31
Chapter 5: Joint Probability Distributions and Random Samples
d. . +(+ =
-#
"#
-"
-#
"#
""
-1#(,(
y K Kxdy y x K dy y x f +=+= ∫ ∫
∞
∞−
= 1#F+" $ !#5, "# ≤ + ≤ -#
e. . y(y is obtained by substitutin) y .or + in (d< /learly .(+,y ≠ . +(+ ⋅ . y(y, so X and Y are
not independent!
10.
a. .(+,y =#
1
otherwise
y x 5,5 ≤≤≤≤
sin/e . +(+ = 1, . y(y = 1 .or 5 ≤ + ≤ , 5 ≤ y ≤
b. P(5!"5 ≤ X ≤ 5!'5, 5!"5 ≤ Y ≤ 5!'5 = P(5!"5 ≤ X ≤ 5!'5 ⋅ P(5!"5 ≤ Y ≤ 5!'5 = (by
independen/e (!5(!5 = !"5
c.
P((X,Y ∈ & = ∫∫ A
dxdy1
= area o. & = 1 > (area o. $ area o.
= -#!-
11
-
"51 ==−
11.
a. p(+,y =GG y
e
x
e y x µ λ µ λ −−
⋅ .or + = #, 1, ", ;< y = #, 1, ", ;
b. p(#,# $ p(#,1 $ p(1,# = [ ] µ λ µ λ ++−− 1e
c. P( X$Y=m = ∑∑=
=−−
= −=−==
m
k
k mk m
k k mk ek mY k X P
## G(G,( µ λ µ λ
G
(
G
(
#
(
m
e
k
m
m
e mm
k
k mk µ λ µ λ µ λ µ λ +
=
+−
=
−+−
∑, so the total H o. errors X$Y also has a
Poisson distribution *ith parameter µ λ + !
1'4
I
II
I1+= x y I1−= x y
5
5
-
8/9/2019 Chapter 5 000
6/31
Chapter 5: Joint Probability Distributions and Random Samples
12.
a. P(X - = #5#!-- #
1( == ∫ ∫ ∫ ∞
−∞ ∞
+−dxedydx xe
x y x
b. Bhe mar)inal pd. o. X is x y x
edy xe −∞
+− =∫ #1(
.or # ≤ +< that o. Y is
"-
1(
1(
1
ydx xe
y x
+=∫ ∞ +−
.or # ≤ y! t is no* /lear that .(+,y is not the produ/t o.
the mar)inal pd.s, so the t*o r!6s are not independent!
c. P( at least one e+/eeds - = 1 > P(X ≤ - and Y ≤ -
= ∫ ∫ ∫ ∫ −−+− −=−-
#
-
#
-
#
-
#
1( 11 dye xedydx xe xy x y x
= -##!"5!"5!1(1 1"--
#
- =−+=−− −−−−∫ eedxee x x
13.
a. .(+,y = . +(+ ⋅ . y(y =
−−
#
y xe
otherwise
y x #,# ≥≥
b. P(X ≤ 1 and Y ≤ 1 = P(X ≤ 1 ⋅ P(Y ≤ 1 = (1 > eA1 (1 > eA1 = !%##
c. P(X $ Y ≤ " = [ ]∫ ∫ ∫ −−−− −− −=
"
#
"("
#
"
#1 dxeedxdye x x
x y x
= 54%!"1( """
#
" =−−=− −−−−∫ eedxee x
d. P(X $ Y ≤ 1 = [ ] "%!"11 11#
1( =−=− −−−−∫ edxee x x ,so P( 1 ≤ X $ Y ≤ " = P(X $ Y ≤ " > P(X $ Y ≤ 1 = !54% A !"% = !--#
14.
a. P(X1 E t, X" E t, ; , X1# E t = P(X1 E t ; P( X1# E t =1#1( t e λ −−
b. . Ksu//essL = 2.ail be.ore t3, then p = P(su//ess = t e λ −−1 ,
and P(7 su//esses amon) 1# trials =k t t ee
k
k −−−−
1#(11#
λ λ
c. P(e+a/tly 5 .ail = P( 5 o. s .ail and other 5 dont $ P(% o. s .ail, .ails, and other 5
dont = ( ) ( ) ( ) ( ) 5%%5 (11%
4(1
5
4t t t t t t
eeeeee λ µ λ µ λ λ −−−−−− −−
+−
15.
a. (y = P( Y ≤ y = P ?(X1 ≤y ∪ ((X" ≤ y ∩ (X- ≤ y@
= P (X1 ≤ y $ P?(X" ≤ y ∩ (X- ≤ y@ A P?(X1 ≤ y ∩ (X" ≤ y ∩ (X- ≤ y@
10#
-
8/9/2019 Chapter 5 000
7/31
Chapter 5: Joint Probability Distributions and Random Samples
= -" 1(1(1( y y y eee λ λ λ −−− −−−+− .or y ≥ #
b. .(y = ′(y = ( ) ( ) y y y y y eeeee λ λ λ λ λ λ λ λ −−−−− −−−+ "1(-1(" = y y ee
λ λ λ λ -" -% −− − .or y ≥ #
(Y = ( ) λ λ λ λ λ λ λ
-"
-1
"1"-%
#
-" =−
=−⋅∫ ∞ −− dyee y
y y
16.
a. .(+1, +- = ( )∫ ∫ −−∞
∞− −= -1
1
#"-"1"-"1 1,,(
x x
dx x xkxdx x x x f
( )( ) "-1-1
11'" x x x x −−− # ≤ +1, # ≤ +-, +1 $ +- ≤ 1
b. P(X1 $ X- ≤ !5 = ∫ ∫ −
−−−5!
#
5!
#1"
"-1-1
1
1(1('" x
dxdx x x x x
= (a.ter mu/h al)ebra !5-1"5
c. ( ) ( )∫ ∫ −−−== ∞
∞− -"
-1-1--11 11'",((1 dx x x x xdx x x f x f x
5
1
-
1
"
11 -%010 x x x x −+− # ≤ +1 ≤ 1
17.
a. ( ,( Y X P *ithin a /ir/le o. radius ) ∫∫ == A
R dxdy y x f A P ,(("
"5!%
1!!1""
==== ∫∫ R
Aof areadxdy
R A
π π
b.
π π
1
"",
"" "
"
==
≤≤−≤≤−
R
R RY
R R X
R P
c.
101
-
8/9/2019 Chapter 5 000
8/31
Chapter 5: Joint Probability Distributions and Random Samples
π π
""
"",
"" "
"
==
≤≤−≤≤−
R
R RY
R R X
R P
d. ( )"
""
"
"1,(
""
""
R
x Rdy
Rdy y x f x f
x R
x R x
π π
−=== ∫ ∫
−
−−
∞
∞− .or >R ≤ + ≤ R and
similarly .or . Y(y! X and Y are not independent sin/e e!)! . +(!4R = . Y(!4R #, yet.(!4R, !4R = # sin/e (!4R, !4R is outside the /ir/le o. radius R!
18.
a. Py8X(y81 results .rom di6idin) ea/h entry in + = 1 ro* o. the oint probability table by p+(1 = !-%:
"-5-!-%!
#0!
18#(8 == x y P
500"!-%!
"#!181(8 == x y P
1'5!-%!
#!18"(8 == x y P
b. Py8X(+8" is reMuested< to obtain this di6ide ea/h entry in the y = " ro* by p+(" = !5#:
y # 1 "
Py8X(y8" !1" !"0 !#
c. P( Y ≤ 1 8 + = " = Py8X(#8" $ Py8X(18" = !1" $ !"0 = !%#
d. PX8Y(+8" results .rom di6idin) ea/h entry in the y = " /olumn by py(" = !-0:
+ # 1 "
P+8y(+8" !#5" !15'4 !'045
10"
-
8/9/2019 Chapter 5 000
9/31
Chapter 5: Joint Probability Distributions and Random Samples
19.
a.#5!1#
(
(
,(8(
"
""
8 ++
==kx
y xk
x f
y x f x y f
X
X Y "# ≤ y ≤ -#
#5!1#
(8(
"
""
8 ++
=ky
y xk y x f Y X "# ≤ + ≤ -#
=
###,-0#
-k
b. P( Y ≥ "5 8 X = "" = ∫ -#
"58 ""8( dy y f X Y
= ∫ =++-#
"5 "
""
'0-!#5!""(1#
""((dy
k
yk
P( Y ≥ "5 = '5!#5!1#((-#
"5
"-#
"5=+= ∫ ∫ dykydy y f Y
c. ( Y 8 X="" = dyk
yk ydy y f y X Y
#5!""(1#
""((""8(
"
""-#
"#8 +
+⋅=⋅ ∫ ∫
∞
∞−
= "5!-'"41"
( Y" 8 X="" = #"0%#!5"#5!""(1#
""(("
""-#
"#
" =++
⋅∫ dyk yk
y
N(Y8 X = "" = ( Y" 8 X="" > ?( Y 8 X="" @" = 0!"%-4'
20.
a. ( ),(
,,(,8
"1,
-"1
"1-,8
"1
"1- x x f
x x x f x x x f
x x
x x x = *here =,( "1, "1 x x f x x the mar)inal oint
pd. o. (X1, X" = --"1 ,,( dx x x x f ∫ ∞
∞−
b. ( )(
,,(8,
1
-"1
1-"8,
1
1-" x f
x x x f x x x f
x
x x x = *here
∫ ∫ ∞
∞−
∞
∞−= -"-"11 ,,((1 dxdx x x x f x f x
21. or e6ery + and y, . Y8X(y8+ = . y(y, sin/e then .(+,y = . Y8X(y8+ ⋅ . X(+ = . Y(y ⋅ . X(+, as
reMuired!
10-
-
8/9/2019 Chapter 5 000
10/31
Chapter 5: Joint Probability Distributions and Random Samples
Section 5.2
22.
a. ( X $ Y = #"(!##(,(( +=+∑∑ x y
y x p y x
1#!1%#1(!151#(!!!#(!5#( =+++++
b. ?ma+ (X,Y@ = ∑∑ ⋅+ x y
y x p y x ,(ma+(
#!4#1(!15(!!!#(!5(#"(!#( =+++=
23. (X1 > X" = ( )∑ ∑= =
⋅−%
#
-
#
"1"1
1 "
,( x x
x x p x x =
(# > #(!#0 $ (# > 1(!#' $ ; $ (% > -(!# = !15(*hi/h also eMuals (X1 > (X" = 1!'# > 1!55
24. Oet h(X,Y = H o. indi6iduals *ho handle the messa)e!
y
h(+,y 1 " - % 5
1 A " - % - "
" " A " - % -
+ - - " A " - %
% % - " A " -
5 - % - " A "
" - % - " A
Sin/e p(+,y = -#1
.or ea/h possible (+,y, ?h(X,Y@ = 0#!",( -#0%
-#1 ==⋅∑∑
x y
y xh
25. (XY = (X ⋅ (Y = O ⋅ O = O"
26. Re6enue = -X $ 1#Y, so (re6enue = (-X $ 1#Y
%!15",5(-5!!!#,#(#,(1#-(5
#
"
#
=⋅++⋅=⋅+= ∑∑= =
p p y x p y x x y
10%
-
8/9/2019 Chapter 5 000
11/31
-
8/9/2019 Chapter 5 000
12/31
Chapter 5: Joint Probability Distributions and Random Samples
( )
( )
( ) ( ) ( )∫ ∫ ∫ ∫ ∞∞∞∞
+−
+=
+−+
=+
=# "## "# " 1
1
1
1
1
11
1( dy
ydy
ydy
y
ydy
y
y y E , and the
.irst inte)ral is not .inite! Bhusρ
itsel. is unde.ined!
33.
Sin/e (XY = (X ⋅ (Y, Co6(X,Y = (XY > (X ⋅ (Y = (X ⋅ (Y A (X ⋅ (Y =
#, and sin/e Corr(X,Y = y x
Y X Cov
σ σ
,(, then Corr(X,Y = #
34.
a. n the dis/rete /ase, Nar?h(X,Y@ = 2?h(X,Y > (h(X,Y@"3 =
∑∑∑∑ −=− x y x y
Y X h E y x p y xh y x pY X h E y xh """ @,((?@,(,(?,(@,((,(?
*ith ∫∫ repla/in) ∑ ∑ in the /ontinuous /ase!
b. ?h(X,Y@ = ?ma+(X,Y@ = 4!#, and ?h"(X,Y@ = ?(ma+(X,Y"@ = (#"(!#"
$(5"
(!# $ ;$ (15"
(!#1 = 1#5!5, so Nar?ma+(X,Y@ = 1#5!5 > (4!#"
= 1-!-%
35.
a. Co6(aX $ b, /Y $ d = ?(aX $ b(/Y $ d@ > (aX $ b ⋅ (/Y $ d
= ?a/XY $ adX $ b/Y $ bd@ > (a(X $ b(/(Y $ d= a/(XY > a/(X(Y = a/Co6(X,Y
b. Corr(aX $ b, /Y $ d =
((8888
,(
((
,(
Y Var X Var ca
Y X acCov
d cY Var aX Var
d cY aX Cov
⋅⋅=
++++
= Corr(X,Y *hen a and / ha6e the same si)ns!
c. hen a and / di..er in si)n, Corr(aX $ b, /Y $ d = ACorr(X,Y!
36. Co6(X,Y = Co6(X, aX$b = ?X⋅(aX$b@ > (X ⋅(aX$b = a Nar(X,
so Corr(X,Y =((
(
((
(
" X Var a X Var
X aVar
Y Var X Var
X aVar
⋅=
⋅ = 1 i. a #, and >1 i. a E #
10
-
8/9/2019 Chapter 5 000
13/31
Chapter 5: Joint Probability Distributions and Random Samples
Section 5.3
37.
P(+1 !"# !5# !-#
P(+" +" 8 +1 "5 %# 5
!"# "5 !#% !1# !#
!5# %# !1# !"5 !15
!-# 5 !# !15 !#4
a.
x "5 -"!5 %# %5 5"!5 5
( ) x p !#% !"# !"5 !1" !-# !#4
( ) µ ==+++= 5!%%#4(!5!!!"#(!5!-"#%(!"5( x E
b.
s" # 11"!5 -1"!5 0##
P(s" !-0 !"# !-# !1"
(s" = "1"!"5 = σ"
38.
a.
B# # 1 " - %
P(B# !#% !"# !-' !-# !#4
b. µ µ ⋅=== ""!"( ## ! E !
c."""
#
"
#
" "40!"!"(0"!5((#
σ σ ⋅==−=−= ! E ! E !
10'
-
8/9/2019 Chapter 5 000
14/31
Chapter 5: Joint Probability Distributions and Random Samples
39.
+ # 1 " - % 5 ' 0 4 1#
+In # !1 !" !- !% !5 ! !' !0 !4 1!#
p(+In !### !### !### !##1 !##5 !#"' !#00 !"#1 !-#" !"4 !1#'
X is a binomial random 6ariable *ith p = !0!
40.
a. Possible 6alues o. Q are: #, 5, 1#! Q = # *hen all - en6elopes /ontain # money, hen/e p(Q = # = (!5- = !1"5! Q = 1# *hen there is a sin)le en6elope *ith 1#, hen/e p(Q =1# = 1 > p(no en6elopes *ith 1# = 1 > (!0- = !%00! p(Q = 5 = 1 > ?!1"5 $ !%00@ = !-0'!
Q # 5 1#
p(Q !1"5 !-0' !%00
&n alternati6e solution *ould be to list all "' possible /ombinations usin) a tree dia)ramand /omputin) probabilities dire/tly .rom the tree!
b. Bhe statisti/ o. interest is Q, the ma+imum o. +1, +", or +-, so that Q = #, 5, or 1#! Bhe population distribution is a s .ollo*s:
+ # 5 1#
p(+ 1I" -I1# 1I5
rite a /omputer pro)ram to )enerate the di)its # > 4 .rom a uni.orm distribution!&ssi)n a 6alue o. # to the di)its # > %, a 6alue o. 5 to di)its 5 > ', and a 6alue o. 1# todi)its 0 and 4! 9enerate samples o. in/reasin) sies, 7eepin) the number o. repli/ations/onstant and /ompute Q .rom ea/h sample! &s n, the sample sie, in/reases, p(Q = #)oes to ero, p(Q = 1# )oes to one! urthermore, p(Q = 5 )oes to ero, but at a slo*errate than p(Q = #!
100
-
8/9/2019 Chapter 5 000
15/31
Chapter 5: Joint Probability Distributions and Random Samples
41.
Tut/ome 1,1 1," 1,- 1,% ",1 "," ",- ",%
Probability !1 !1" !#0 !#% !1" !#4 !# !#-
x 1 1!5 " "!5 1!5 " "!5 -
r # 1 " - 1 # 1 "
Tut/ome -,1 -," -,- -,% %,1 %," %,- %,%
Probability !#0 !# !#% !#" !#% !#- !#" !#1
x " "!5 - -!5 "!5 - -!5 %
r " 1 # 1 - " 1 "
a.
x 1 1!5 " "!5 - -!5 %
( ) x p !1 !"% !"5 !"# !1# !#% !#1
b. P ( )5!"≤ x = !0
c.
r # 1 " -
p(r !-# !%# !"" !#0
d. 5!1( ≤ X P = P(1,1,1,1 $ P(",1,1,1 $ ; $ P(1,1,1," $ P(1,1,"," $ ; $ P(",",1,1$ P(-,1,1,1 $ ; $ P(1,1,1,-= (!%% $ %(!%-(!- $ (!%"(!-" $ %(!%"(!"" = !"%##
42.
a.
x "'!'5 "0!# "4!' "4!45 -1!5 -1!4 --!
( ) x p-#%
-#"
-#
-#%
-#0
-#%
-#"
b.
x "'!'5 -1!5 -1!4
( ) x p-1
-1
-1
c. all three 6alues are the same: -#!%---
104
-
8/9/2019 Chapter 5 000
16/31
655545352515
70
60
50
40
30
20
10
0
F r e q u e n c y
P-Value: 0.000
A-Squared: 4.428
Anderson-Darln! "or# al$y%es$
6050403020
.&&&
.&&
.&5
.80
.50
.20
.05
.01
.001
P r o ' a ' l $ y
"or#al Pro'a'l$y Plo$
Chapter 5: Joint Probability Distributions and Random Samples
43. Bhe statisti/ o. interest is the .ourth spread, or the di..eren/e bet*een the medians o. theupper and lo*er hal6es o. the data! Bhe population distribution is uni.orm *ith & = 0 and =1#! Use a /omputer to )enerate samples o. sies n = 5, 1#, "#, and -# .rom a uni.ormdistribution *ith & = 0 and = 1#! Feep the number o. repli/ations the same (say 5##, .ore+ample! or ea/h sample, /ompute the upper and lo*er .ourth, then /ompute thedi..eren/e! Plot the samplin) distributions on separate histo)rams .or n = 5, 1#, "#, and -#!
44. Use a /omputer to )enerate samples o. sies n = 5, 1#, "#, and -# .rom a eibull distribution*ith parameters as )i6en, 7eepin) the number o. repli/ations the same, as in problem %-abo6e! or ea/h sample, /al/ulate the mean! elo* is a histo)ram, and a normal probability
plot .or the samplin) distribution o. x .or n = 5, both )enerated by Qinitab! Bhis samplin)distribution appears to be normal, so sin/e lar)er sample sies *ill produ/e distributions thatare /loser to normal, the others *ill also appear normal!
45. Usin) Qinitab to )enerate the ne/essary samplin) distribution, *e /an see that as n in/reases,the distribution slo*ly mo6es to*ard normality! Vo*e6er, e6en the samplin) distribution .orn = 5# is not yet appro+imately normal!n = 1#
n = 5#
14#
0 10 20 30 40 50 60 70 80 &0
0
10
20
30
40
50
60
70
80
&0
F r e q u e n c y
P-Value: 0.000
A-Squared: 7.406
Anderson-Darln! "or#al$y %es$
85756555453525155
.&&&
.&&
.&5
.80
.50
.20
.05
.01
.001
P r o ' a ' l $ y
n(10
"or#al Pro'a'l$y Plo$
-
8/9/2019 Chapter 5 000
17/31
Chapter 5: Joint Probability Distributions and Random Samples
Section 5.4
46. µ = 1" /m σ = !#% /m
a. n = 1 cm X E 1"( == µ cmn
x
x #1!
%
#%!===
σ σ
b. n = % cm X E 1"( == µ cmn
x
x ##5!0
#%!===
σ σ
c. X is more li7ely to be *ithin !#1 /m o. the mean (1" /m *ith the se/ond, lar)er,
sample! Bhis is due to the de/reased 6ariability o. X *ith a lar)er sample sie!
47. µ = 1" /m σ = !#% /m
a. n = 1 P( 11!44 ≤ X ≤ 1"!#1 =
−≤≤−
#1!
1"#1!1"
#1!
1"44!11 " P
= P(A1 ≤ W ≤ 1
= Φ(1 A Φ(A1
=!0%1- A !150'=!0"
b. n = "5 P( X 1"!#1 =
−>
5I#%!
1"#1!1" " P = P( W 1!"5
= 1 A Φ(1!"5
= 1 A !04%%=!1#5
48.
a. 5#== µ µ X , 1#!1##
1===
n
x
x
σ σ
P( %4!'5 ≤ X ≤ 5#!"5 =
−≤≤
−1#!
5#"5!5#
1#!
5#'5!%4 " P
= P(A"!5 ≤ W ≤ "!5 = !40'
b. P( %4!'5 ≤ X ≤ 5#!"5 ≈
−≤≤
−1#!
0!%4"5!5#
1#!
0!%4'5!%4 " P
= P(A!5 ≤ W ≤ %!5 = !415
141
-
8/9/2019 Chapter 5 000
18/31
-
8/9/2019 Chapter 5 000
19/31
Chapter 5: Joint Probability Distributions and Random Samples
53. µ = 5#, σ = 1!"
a. n = 4
P( X ≥ 51 = ##"!44-0!15!"(4I"!1
5#51 =−=≥=
−≥ " P " P
b. n = %#
P( X ≥ 51 = #"'!5(%#I"!1
5#51 ≈≥=
−≥ " P " P
54.
a. 5!"== µ µ X , 1'!5
05!===
n
x
x
σ σ
P( X ≤ -!##= 40#-!#!"(1'!
5!"##!-=≤=
−≤ " P " P
P("!5 ≤ X ≤ -!##= %0#-!5!"(##!-( =≤−≤= X P X P
b. P( X ≤ -!##= 44!I05!
5!"##!-=
−≤
n " P implies that ,--!"
I05
-5!=
n .rom
*hi/h n = -"!#"! Bhus n = -- *ill su..i/e!
55. "#== np µ %%!-== np#σ
a. P( "5 ≤ X ≈ #40!-#!1(%%!-
"#5!"%=≤=
≤
− " P " P
b. P( 15 ≤ X ≤ "5 ≈
−≤≤−
%%!-
"#5!"5
%%!-
"#5!1% " P
000"!54!154!1( =≤≤−= " P
56.
a. ith Y = H o. ti/7ets, Y has appro+imately a normal distribution *ith 5#== λ µ ,
#'1!'== λ σ , so P( -5 ≤ Y ≤ '# ≈
−≤≤
−#'1!'
5#5!'#
#'1!'
5#5!-% " P = P( A"!14
≤ W ≤ "!4# = !40-0
b. Vere "5#= µ , 011!15,"5#" == σ σ , so P( ""5 ≤ Y ≤ "'5 ≈
−≤≤−
011!15
"5#5!"'5
011!15
"5#5!""% " P = P( A1!1 ≤ W ≤ 1!1 = !04"
57. (X = 1##, Nar(X = "##, 1%!1%= xσ , so P( X ≤ 1"5 ≈
−≤
1%!1%
1##1"5 " P
= P( W ≤ 1!'' = !41
14-
-
8/9/2019 Chapter 5 000
20/31
-
8/9/2019 Chapter 5 000
21/31
Chapter 5: Joint Probability Distributions and Random Samples
d. ( X1 $ X" $ X- = 15#, N(X1 $ X" $ X- = -, -"1
=++ x x xσ
P(X1 $ X" $ X- ≤ "## = 45"5!'!1(
15#1#=≤=
−≤ " P " P
e *ant P( X1 $ X" ≥ "X-, or *ritten another *ay, P( X1 $ X" A "X-≥ #
( X1 $ X" A "X- = %# $ 5# > "(# = A-#,
N(X1 $ X" A "X- = ,'0% "-"""1 =++ σ σ σ -, sd = 0!0-", so
P( X1 $ X" A "X-≥ # = ###-!%#!-(0-"!0
-#(#=≥=
−−≥ " P " P
60. Y is normally distributed *ith ( ) ( ) 1-
1
"
15%-"1 −=++−+= µ µ µ µ µ µ Y , and
''45!1,1'!-4
1
4
1
4
1
%
1
%
1 "5
"
%
"
-
"
"
"
1
" ==++++= Y Y
σ σ σ σ σ σ σ !
Bhus, ( ) "0''!5(!''45!1
1(## =≤=
≤−−
=≤ " P " P Y P and
( ) -0!1"!1#(''45!1
"#11 =≤≤=
≤≤=≤≤− " P " P Y P
61.
a. Bhe mar)inal pm.s o. X and Y are )i6en in the solution to +er/ise ', .rom *hi/h (X= "!0, (Y = !', N(X = 1!, N(Y = !1! Bhus (X$Y = (X $ (Y = -!5, N(X$Y= N(X $ N(Y = "!"', and the standard de6iation o. X $ Y is 1!51
b. (-X$1#Y = -(X $ 1#(Y = 15!%, N(-X$1#Y = 4N(X $ 1##N(Y = '5!4%, and thestandard de6iation o. re6enue is 0!'1
62. ( X1 $ X" $ X- = ( X1 $ (X" $ (X- = 15 $ -# $ "# = 5 min!,
N(X1 $ X" $ X- = 1" $ "" $ 1!5" = '!"5, 4"!""5!'
-"1==++ x x xσ
Bhus, P(X1 $ X" $ X- ≤ # = #-1%!0!1(4"!"
5#=−≤=
−≤ " P " P
63.
a. (X1 = 1!'#, (X" = 1!55, (X1X" =--!-,(
1 "
"1"1 =∑∑ x x
x x p x x, so
Co6(X1,X" = (X1X" A (X1 (X" = -!-- > "!-5 = !45
b. N(X1 $ X" = N(X1 $ N(X" $ " Co6(X1,X"= 1!54 $ 1!#0'5 $ "(!45 = %!#'5
145
-
8/9/2019 Chapter 5 000
22/31
Chapter 5: Joint Probability Distributions and Random Samples
64. Oet X1, ;, X5 denote mornin) times and X, ;, X1# denote e6enin) times!a. (X1 $ ;$ X1# = (X1 $ ; $ (X1# = 5 (X1 $ 5 (X
= 5(% $ 5(5 = %5
b. Nar(X1 $ ;$ X1# = Nar(X1 $ ; $ Nar(X1# = 5 Nar(X1 $ 5Nar(X
--!0
1"
0"#
1"
1##
1"
%5 ==
+=
c. (X1 > X = (X1 A (X = % > 5 = A1
Nar(X1 > X = Nar(X1 $ Nar(X = '!1-1"
1%
1"
1##
1"
%==+
d. ?(X1 $ ; $ X5 > (X $ ; $ X1#@ = 5(% > 5(5 = A5Nar?(X1 $ ; $ X5 > (X $ ; $ X1#@
= Nar(X1 $ ; $ X5 $ Nar(X $ ; $ X1#@ = 0!--
65. µ = 5!##, σ = !"
a.
-
8/9/2019 Chapter 5 000
23/31
Chapter 5: Joint Probability Distributions and Random Samples
= !"5 $ "(5(1#(A!"5 $ 1## = 01!"5
67. Oettin) X1, X", and X- denote the len)ths o. the three pie/es, the total len)th isX1 $ X" A X-! Bhis has a normal distribution *ith mean 6alue "# $ 15 > 1 = -%, 6arian/e !"5$!1$!#1 = !%", and standard de6iation !%01! Standardiin) )i6es
P(-%!5 ≤ X1 $ X" A X- ≤ -5 = P(!'' ≤ W ≤ 1!5% = !1500
68. Oet X1 and X" denote the (/onstant speeds o. the t*o planes!a. &.ter t*o hours, the planes ha6e tra6eled "X1 7m! and "X" 7m!, respe/ti6ely, so the
se/ond *ill not ha6e /au)ht the .irst i. "X1 $ 1# "X", i!e! i. X" > X1 E 5! X" > X1 has amean 5## > 5"# = A"#, 6arian/e 1## $ 1## = "##, and standard de6iation 1%!1%! Bhus,
!41!''!1(1%!1%
"#(55( 1" =1# ≤ "X" > 1# > "X1 ≤ 1#,
i!e! # ≤ X" > X1 ≤ 1#! Bhe /orrespondin) probability is
P(# ≤ X" > X1 ≤ 1# = P(1!%1 ≤ W ≤ "!1" = !40-# A !4"#' = !#"-!
69.
a. (X1 $ X" $ X- = 0## $ 1### $ ## = "%##!
b. &ssumin) independen/e o. X1, X" , X-, Nar(X1 $ X" $ X-= (1" $ ("5" $ (10" = 1"!#5
c. (X1 $ X" $ X- = "%## as be.ore, but no* Nar(X1 $ X" $ X-= Nar(X1 $ Nar(X" $ Nar(X- $ "Co6(X1,X" $ "Co6(X1, X- $ "Co6(X", X- = 1'%5,*ith sd = %1!''
70.
a. ,5!( =iY E so%
1(5!((
11
+==⋅= ∑∑
==
nniY E i% E
n
i
n
i
i
b. ,"5!( =iY Var so"%
1"(1("5!((
1
"
1
" ++==⋅= ∑∑==
nnniY Var i% Var
n
i
n
i
i
14'
-
8/9/2019 Chapter 5 000
24/31
Chapter 5: Joint Probability Distributions and Random Samples
71.
a. ,'"""111"
#""11 % X a X a xdx% X a X a $ ++=++= ∫ so
(Q = (5(" $ (1#(% $ ('"(1!5 = 150m
( ) ( ) ( ) ( ) ( ) ( ) "5!%-#"5!'"11#5!5 """"""" =++= $ σ , '%!"#= $ σ
b. 4'00!#-!"('%!"#
150"##"##( =≤=
−≤=≤ " P " P $ P
72. Bhe total elapsed time bet*een lea6in) and returnin) is Bo = X1 $ X" $ X- $ X%, *ith
,%#( =o! E %#" =
o! σ , %''!5=
o! σ ! Bo is normally distributed, and the desired
6alue t is the 44th per/entile o. the lapsed time distribution added to 1# &!Q!: 1#:## $?%#$(5!%''("!--@ = 1#:5"!'
73.
a. oth appro+imately normal by the C!O!B!
b. Bhe di..eren/e o. t*o r!6!s is ust a spe/ial linear /ombination, and a linear /ombination
o. normal r!6s has a normal distribution, so Y X − has appro+imately a normal
distribution *ith 5=−Y X µ and "1!1,"4!"-5
%#
0""
" ==+= −− Y X Y X σ σ
c. ( )
−≤≤
−−≈≤−≤−
"1-!1
51
"1-!1
5111 " P Y X P
##0!%'!"'#!-( ≈−≤≤−= " P
d.
( )!##1#!#0!-(
"1-!1
51#1#
=≥=
−
≥≈≥− " P " P Y X P
Bhis probability is
Muite small, so su/h an o//urren/e is unli7ely i. 5"1 =− µ µ , and *e *ould thus doubtthis /laim!
74. X is appro+imately normal *ith -5'(!5#(1 == µ and 5!1#-(!'(!5#("
1 ==σ ,as is Y *ith -#" = µ and 1"
"
" =σ ! Bhus 5=−Y X µ and 5!""" =−Y X σ , so
( ) %0"!#11!"('%!%
#
'%!%
1#55 =≤≤−=
≤≤
−≈≤−≤− " P " P Y X p
140
-
8/9/2019 Chapter 5 000
25/31
Chapter 5: Joint Probability Distributions and Random Samples
Supplementary Exercises
75.
a. pX(+ is obtained by addin) oint probabilities a/ross the ro* labeled +, resultin) in pX(+= !", !5, !- .or + = 1", 15, "# respe/ti6ely! Similarly, .rom /olumn sums py(y = !1, !-5, !55 .or y = 1", 15, "# respe/ti6ely!
b. P(X ≤ 15 and Y ≤ 15 = p(1",1" $ p(1",15 $ p(15,1" $ p(15,15 = !"5
c. p+(1" ⋅ py(1" = (!"(!1 ≠ !#5 = p(1",1", so X and Y are not independent! (&lmost any
other (+,y pair yields the same /on/lusion!
d. -5!--,((( =+=+ ∑ ∑ y x p y xY X E (or = (X $ (Y = --!-5
e. 05!-,(( =+=− ∑ ∑ y x p y xY X E
76. Bhe rollAup pro/edure is not 6alid .or the '5th per/entile unless #1 =σ or #" =σ or both
1σ and #" =σ , as des/ribed belo*!Sum o. per/entiles: (((( "1"1""11 σ σ µ µ σ µ σ µ +++=+++ " " "
Per/entile o. sums: """1 "1
( σ σ µ µ +++ " Bhese are eMual *hen W = # (i!e! .or the median or in the unusual /ase *hen
""
"1 "1σ σ σ σ +=+ , *hi/h happens *hen #1 =σ or #" =σ or both 1σ and #" =σ !
77.
a. ∫ ∫ ∫ ∫ ∫ ∫ −−
−
∞
∞−
∞
∞−+==
-#
"#
-#
#
"#
#
-#
"#,(1
x x
xkxydydxkxydydxdxdy y x f
"5#,01
-
-
"5#,01=⇒⋅= k k
b.
+−=
−==
∫ ∫
−
−
−
-#%5#(
1#"5#((
-
"1"
-#
#
"-#
"#
x x xk kxydy
x xk kxydy x f x
x
x X -#"#
"##
≤≤≤≤
x
x
and by symmetry . Y(y is obtained by substitutin) y .or + in . X(+! Sin/e . X("5 #, and. Y("5 #, but .("5, "5 = # , . X(+ ⋅ . Y(y ≠ .(+,y .or all +,y so X and Y are not
independent!
c. ∫ ∫ ∫ ∫ −−
− +=≤+
"5
"#
"5
#
"#
#
"5
"#"5(
x x
xkxydydxkxydydxY X P
144
-#=+ y x
"#=+ y x
-
8/9/2019 Chapter 5 000
26/31
Chapter 5: Joint Probability Distributions and Random Samples
-55!"%
"5,"-#
"5#,01
- =⋅=
d. ( )∫ −⋅=+=+"#
#
"1#"5#"((( dx x xk xY E X E Y X E
( )∫ +−⋅+-#
"#
-
"1"-#%5# dx x x xk x 44!"5'!,-51(" == k
e. ∫ ∫ ∫ ∫ −
−
∞
∞−
∞
∞− =⋅=
"#
#
-#
"#
"",(( x
xdydx ykxdxdy y x f xy XY E
%1#-!1--
###,"5#,--
-
-#
"#
-#
#
"" =⋅=+ ∫ ∫ − k
dydx ykx x
, so
Co6(X,Y = 1-!%1#- > (1"!40%5" = A-"!14, and (X" = (Y" = "#%!15%, so
#10"!-40%5!1"(15%!"#% """ =−== y x σ σ and 04%!#10"!-
14!-"−=
−= ρ
f. Nar (X $ Y = Nar(X $ Nar(Y $ "Co6(X,Y = '!
78. Y(y = P( ma+(X1, ;, Xn ≤ y = P( X1 ≤ y, ;, Xn ≤ y = ?P(X1 ≤ y@n
n y
−=
1##
1## .or
1## ≤ y ≤ "##!
Bhus . Y(y = ( )1
1##1##
−− nn
yn
.or 1## ≤ y ≤ "##!
( ) ( )∫ ∫ −− +=−⋅=1##
#
1"##
1##
11##
1##1##
1##( duuu
ndy y
n yY E n
n
n
n
1##1
1"
11##1##
1##1##
1##
#⋅
+
+=
+
+=+= ∫ nn
n
nduu
n nn
79. -%##"###4##5##( =++=++ " Y X E
#1%!1"--5
10#
-5
1##
-5
5#(
"""
=++=++ " Y X Var , and the std de6 = 11!#4!
1#!4(-5##( ≈≤=≤++ " P " Y X P
"##
-
8/9/2019 Chapter 5 000
27/31
-
8/9/2019 Chapter 5 000
28/31
Chapter 5: Joint Probability Distributions and Random Samples
87.
a. !",("(""""""
yY X x y x aaY X aCovaY aX Var σ ρ σ σ σ σ σ ++=++=+
Substitutin)
X
Y a
σ
σ = yields ( ) #1"" """" ≥−=++ ρ σ σ ρ σ σ Y Y Y Y , so 1−≥ ρ
b. Same ar)ument as in a
c. Suppose 1= ρ ! Bhen ( ) ( ) #1" " =−=− ρ σ Y Y aX Var , *hi/h implies thatk Y aX =− (a /onstant, so k aX Y aX −=− , *hi/h is o. the .orm aX + !
88. ∫ ∫ ⋅−+=−+1
#
1
#
"" !,((( dxdy y x f t y xt Y X E Bo .ind the minimiin) 6alue o. t,
ta7e the deri6ati6e *ith respe/t to t and eMuate it to #:
t dxdy y xtf y x f t y x =⇒=−−+= ∫ ∫ ∫ ∫ 1
#
1
#
1
#
1
#,(#,(1(("#
(,((1
#
1
#Y X E dxdy y x f y x +=⋅+= ∫ ∫ , so the best predi/tion is the indi6iduals
e+pe/ted s/ore ( = 1!1'!
89.
a. ith Y = X1 $ X",
( ) ( ) ( ) 1""
1"
"
1"
"
"I# #1
"I
"1"1
1"1
1
1 "I"
1
"I"
1
dxdxe x x y '
x x y x y
Y
⋅Γ
⋅Γ=
+−−−−
∫ ∫
ν ν
ν ν
ν ν
! ut the inner
inte)ral /an be sho*n to be eMual to ( ) ( )( ) "I1@"I?
"1
"I
"1
"1 "I("
1 ye y
−−++ +Γ
ν ν
ν ν ν ν
,
.rom *hi/h the result .ollo*s!
b. y a,"
"
"
1 " " + is /hiAsMuared *ith "=ν , so ( )"
-
"
"
"
1 " " " ++ is /hiAsMuared *ith-=ν , et/, until ""1 !!! n " " ++ 4s /hiAsMuared *ith n=ν
"#"
-
8/9/2019 Chapter 5 000
29/31
Chapter 5: Joint Probability Distributions and Random Samples
c.σ
µ −i X is standard normal, so"
−
σ
µ i X is /hiAsMuared *ith 1=ν , so the sum is
/hiAsMuared *ith n=ν !
"#-
-
8/9/2019 Chapter 5 000
30/31
Chapter 5: Joint Probability Distributions and Random Samples
90.
a. Co6(X, Y $ W = ?X(Y $ W@ > (X ⋅ (Y $ W
= (XY $ (XW > (X ⋅ (Y > (X ⋅ (W
= (XY > (X ⋅ (Y $ (XW > (X ⋅ (W
= Co6(X,Y $ Co6(X,W!
b. Co6(X1 $ X" , Y1 $ Y" = Co6(X1 , Y1 $ Co6(X1 ,Y" $ Co6(X" , Y1 $ Co6(X" ,Y"(apply a t*i/e = 1!
91.
a. (((( """"
11 X V E % V E % V X V E % =+=+=+= σ σ and
++=++= ,(,(,(,( ""1"1 E % Cov% % Cov E % E % Cov X X Cov"
"11 (,(,(,( w% V % % Cov E E Cov% E Cov σ ===+ !
Bhus, ""
"
""""
"
E %
%
E % E %
%
σ σ
σ
σ σ σ σ
σ ρ
+=
+⋅+=
b. 4444!###1!1
1=
+= ρ
92.
a. Co6(X,Y = Co6(&$D, $= Co6(&, $ Co6(D, $ Co6(&, $ Co6(D,= Co6(&,! Bhus
""""
,(,(
E ( ) A
( ACovY X Corr
σ σ σ σ +⋅+=
""""
,(
E (
(
) A
A
( A
( ACov
σ σ
σ
σ σ
σ
σ σ +⋅
+⋅=
Bhe .irst .a/tor in this e+pression is Corr(&,, and (by the result o. e+er/ise '#a the
se/ond and third .a/tors are the sMuare roots o. Corr(X1, X" and Corr(Y1, Y",respe/ti6ely! Clearly, measurement error redu/es the /orrelation, sin/e both sMuareAroot.a/tors are bet*een # and 1!
b. 055!4#"5!01##! =⋅ ! Bhis is disturbin), be/ause measurement error substantiallyredu/es the /orrelation!
"#%
-
8/9/2019 Chapter 5 000
31/31
Chapter 5: Joint Probability Distributions and Random Samples
93. [ ] "1"#,,,(("#1
151
1#1
%-"1 =++== µ µ µ µ hY E
Bhe partial deri6ati6es o. ,,,( %-"1 µ µ µ µ h *ith respe/t to +1, +", +-, and +% are ,"1
%
x
x−
,""
%
x x− ,"
-
%
x x− and
-"1
111 x x x ++
, respe/ti6ely! Substitutin) +1 = 1#, +" = 15, +- = "#,
and +% = 1"# )i6es >1!", A!5---, A!-###, and !"1', respe/ti6ely, so N(Y = (1(A1!"" $ (1
(A!5---" $ (1!5(A!-###" $ (%!#(!"1'" = "!'0-, and the appro+imate sd o. y is 1!%!
94. Bhe .our se/ond order partials are ,"
-
1
%
x
x,
"-
"
%
x
x,
"-
-
%
x
xand # respe/ti6ely! Substitution
)i6es (Y = " $ !1"## $ !#-5 $ !#--0 = "!104%!
"#5