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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(",-=!%

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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!

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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

"#

"#

-#

-#

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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

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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@

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= -" 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.

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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

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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!

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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

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( )

( )

( ) ( ) ( )∫ ∫ ∫ ∫ ∞∞∞∞

+−

+=

+−+

=+

=# "## "# " 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 #

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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 !

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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 = #!

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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: -#!%---

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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$y Plo$


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$y Plo$

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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

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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-

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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

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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.

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= !"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'

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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

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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

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-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

"##

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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

"#"

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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 !

"#-

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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!

"#%

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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

chapter 5 000

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