EASTERN MEDITERRANEAN UNIVERSITY
DEPARTMENT OF MATH EMATICS
Mark
Question
1.
- Probability and Statisticat Methods
2OL2 - 2OL3 Spring Semester
Math322
a
Final Exam
Date: May 31, 2AL3; Duration;12O min.; Note: Calculator is
allowed.
3.
4.
5.
1. Part A, (5pts,) How many different
car registration plates can be
made consisting of the letters A,B,C, and the digits I,2,3,4 where
the three letters appear before the four numbers and each letter
and digit can only appear once in any registration plate?
6.
7.
L
1
3l , 4 {
-:6.".e/y._irllt,
TOTAL
Part B. (Spts.) In how many ways can
are together?
1l(
I
where the L's
-- ls':
l,
,-€i l,. dnd E b€ twc !nc*pendena evcnts-yJiin ?iA'
(a) (apts.) P(B)=
Li:
- I (t t,')E )Jt'}i) -- fcA):. f'' ( A) -+ 7(t) i'(tt)i,(i., )
['fr;)- :)/ - n
s.'i;if;(
l( ,tvi:-\)-'71x) -- f '( r-)
'
;i,i
(b)(apts.) P(Alir)=
_
i, (A)
\,,v. l
.
L;.-.r
3.r,1cp
c{
cttt
tl
l
;G;-+l
a
\
/
oI{!
rr_,-(_
{1)
1"i,r-,,
t,L.t
-y'-
t-
(i^'.,J .r,i,
i,r,{rpc,.",.
lid
3, A random sample of 100 people were asked whether they ate healthy foods and whether
exercised three or more times per week. The results are shown in the following Venn diagrt
Suppose that one of these individuals was chosen at random. Fill in the following tablJ ar
calculate the probability of each of the possibilities given in (a)-(e)
"B"Exercised 3
or mor.e tlmes
per,"veek
Did not
exercise 3 er
more times per
"A"Ate
Did n,ot eat
Healthy
healthy
Foods
foods
*r
xtlr
LI
4+
ureek
t
Totel
' .:_
lisr;rrireil
tltpa:r1 ot, tt11lre
til6r:r Fsl. rgr'rrli
lt,\*
Ll)
:)
otsl
c._
:)--)
;trtg
futillffu,,i
Iirtrls
I
i
(spts.)
(a)(2pts.) The person ate healthy foods.
r'(4.4x1 ril
(b)(2pts.) The person exercised three of more times per week.
,-(F+
)
''/,
t,s
(c){2pts.} The person ate healthy foods and exercised three or more times per week.
Pc/\ o it)
,t;t
*,
, *Ld::.
- *7t w
:*lg:: 4ii ir*i *:i-
"flr)-t*
i-t/
/(-
i',t:;i!! r-r'r t-::a;j:.
t'(*)= *%u,
(e) (2pts.) The person exercised three or more times per week,
r( El fr)_
4.
given that they ate healthy foods.
)J/tlv" d),1$J'i
/
Let X denote the am ount of time a book on two-hour reserve is actually checked out, and suppose
the cumulative distri bution function is given as follows:
x<0
fo,
lz
lxI
F(x) = i-
I 16
l4
[*
.
0<x<4
I
x>4
2
J
,3. (lopts')
Z is the standard normal random variable. Find the value of
a, if P(o.Z <I.12)=9.6g45..
o,(
-*r{
7'
Consider the following discrete joint probability distribution of X and
Y. .-Q;=
Y
-4
2
7
g(x)
1/4
r/a
1/z
5
r/a
r/+
UA
7/e
7/z
h(v)
3/B
3/B
1/+
1
X
1
Find the,
(a) expectation of X; E(X). (3pts.)
trrA:,a
4 !-sj_=[=_\
4,*)-x=tif
ib)
expectatiori cf y', i_tY)" {3pts.}
e{>r):> ,4&q)== -o,,*rt{u
r",i
trl
1=''t,tr+
(c) joint expectation of X and Y; E(XY). (4pts.)
(e) correlation coefficient of X and Y; p(X,Y), and identify it. (4pts.)
8
0"o(
'*€&1 -(t, Cx)
xx )
"r=
15I 6?
A *'+,/
-)tV_ /''!^ \--
I
5;,.=
6-..r'
n,f4
(:
1"R
t rz7+r,if
use the cumurative distribution
function (cdf) to obtain
the forowing.
(a) (3pts.) p(X z)
=
7
(x<))
=_
(b)
(3pts.) P(3<x
fG)
,'= D,{
=
=o'y#
(c) (3pts,)
Find the probabitity
density function f(x).
k^=2' % P<x< +
o ,, , b1,ul
S.r./pt,v,r
-eo"{kh@,,
(c) (1Opts.) What is the
minimum salary of the
top o/olO of the employees?
T&r^)-=
Dro
r
qr_
!
Useful Formulae and Tables i
i.......
(n\(u-"\
(:),
n' r,x
[,J[,-, ,
-l'l--
=0,t,2,
r("')-[a(x)l',
J
e(.ry)- n(x)a(r)
\,)
x-p
Cov(X,Y)
OxOy
,
^lrpq ,
npq
6
-tt1P
Aleas unclei the Normal Curve
0.o 0.5000
0.1 0.5398
o.2 0.5793
0.3 0.6179
o.4 0.6554
0.5 0.6915
0.6 0.7257
0"7 0.7580
O.E U.7EE i
el..{*
i_}.815!,
n.{} {1.84i3
i:1
-L-t64:::
'i
"'.'.: :i11j...j.
: "::,
:.4
1.5
0.5040
0.5438
0.5832
0.6217
0"6591
0.6950
0.729L
0.76i
1
0.79iL)
t
3i86
*.8+38
?.840,.r
: !:t,tii:,.
:i{14!
ti.::)\i,:...
,,
u.9i9"r-l
ii.920i'
0.9332
0.9345
0.9463
1.6*: o,3aBF
t.7 0.9554
1.8 0.9641
1.9 0.9713
2.O 0.9772
2.1, 0.9821
2.2 0.9861
2.3 0.9893
2"4 0.9918
2.5 0.9938
2.6 0.9953
2"7 0.9965
2.8 0.9974
2.9 0.9981
3.0 0.9987
3.1 0.9990
3.2 0.9993
3.3 0.9995
3.4 0.9997
.
.o5
.03
.01
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0"9993
0.9995
0.9997
0.5080 0.5120
0.5478 0.5517
0.5871 0.5910
0.6255 0.6293
0.6628 0.6664
0.6985 0.7019
0.7324 0.7357
0.7642 0.7673
0"?93!i +.79e:
0.5160
0.5557
0.5948
0.5199
0.5596
0.633r
0.6368
0.6736
0.6700
0.7054
0.7389
a.77A4
ij.799i
0,5987
0.7088
0,7422
A.7734
0.8023
a,,.8).;
.i ,i:,l::
.'.8ti-
ii.846i
t.ii,18r
ii.850al i;.35-::
o.B?08
0.87?i'
t-tre6€6
:;2:-}itlA{€
r;.:!r,it'f, -:.il{:1.,
i).3222 ,i.923tj
0.9357 0.9370
0.9474 0.9484
0.9573 0.9582
0.9656 0.9664
0.9726 0.9732
0.9783 0.9788
0.9830 0.9834
0.9868 0.9871
0.9898 0.9901
0.9922 0.9925
0.9941 0.9943
0.9956 0.9957
8.9967 0.9968
0.9976 0.9977
0.9982 0.9983
0.9987 0.9988
0.9991 0.9991
0.9994 0.9994
0.9995 0.9996
0.9997 0.9997
ii.82E!)
iJ 8749
il t?ct:
.1
.i.9{}.t-';'
;
l.g?5
,,r..92e45
;
,_()r,
,:.;:,ii
.06
.09
0.5239 0.5279 0.5319 0.5359
0.5636 0.5675 a.57L4 0.5753
0.6026 0.6064 0.6103 0.6141
0.6406 0.6443 0.6480 ii.6517
0.6772 0.6808 0.6844 0.6879
0.7L23 0.7757 0.7190 0.7224
0.7454 0.7486 0.7517 0.7549
0.7764 4.7794 0.7823 0.7852
0.8051 0.8078 0.8i06 0.8133
C.83i-i li.8:i4* 0"8365 ,J.8389
';.8554 0.8577 C.8599 l.8il:i.
0.8770 0.87911 -a.BQC 0,883$
'.:.2t1{;'. '1.8i}8{i
r 39!\
-
;:
0.9382 0.9394
0.9495 0.9505
0.9591 0.9599
0.9671 0.9678
0.9738 0.9744
0.9793 0.9798
0.9838 0.9842
0.9875 0.9878
0.9904 0.9906
0.9927 0.9929
0.9945 0.9946
0.995S 0.9960
0.9969 0.997G
a.9977 0.9978
0.9984 0.9984
0.9988 0.9989
0.9992 0.9992
0.9994 0.9994
0.9996 0.9996
0.9997 0.9997
.o8
.o7
1.' ,.
''
lll:
tlQr
t.iSiia
*.93i5
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9418
0.9525
0.9616
0.9693
0.9756
a.9429
0.9535
0.962t,
0.9699
0.9761
0.9441
0.9808
0.9850
0.98i2
0.9817
0.9857
0.9890
0.9916
0.9936
0.996i
0.997i
0.9962
A"9972
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
0.9979
0.9985
:, CiTi.
r
0.9884
0.9911
0,9932
0.9949
0.9989
0.9992
0.9995
0.9996
0.9997
0.9354
0.9887
0
9913
0.9545
0.9633
0.9706
0.9767
0.9934
0.9951 0.9952
0.9963 0,9964
0.9973 A:9W4
0.9980 0,9981
0.9986 0,9985
0.9990 0,9990
0.9993 0;9993
0.9995 0'9995
0.9996 0.9997
0.9997 0.9998