Q Quest

ions
No.

1.Def
iner
andom v
ari
abl
e.

2.Whatdoy
oumeanbydi
str
ibut
ionofar
andom v
ari
abl
e?

3.Av
ari
abl
ethatcanassumeanypossi
blev
aluebet
weent
wopoi
ntsi
scal
l
ed:

(
a)Di
scr
eter
andom v
ari
abl
e

(
b)Cont
inuousr
andom v
ari
abl
e

(
c)Di
scr
etesampl
espace

(
d)Random v
ari
abl
e

4.Af
ormul
aorequat
ionusedt
orepr
esentt
hepr
obabi
l
ity
'di
str
ibut
ionofacont
inuousr
andom v
ari
abl
eiscal
l
ed:

(
a)Pr
obabi
l
itydi
str
ibut
ion

(
b)Di
str
ibut
ionf
unct
ion

(
c)Pr
obabi
l
itydensi
tyf
unct
ion

(
d)Mat
hemat
ical
expect
ati
on

5.I
fXi
sAdi
scr
eter
andom v
ari
abl
eandf
(x)i
sthepr
obabi
l
ityofX,
thent
heexpect
edv
alueoft
hisr
andom v
ari
abl
eisequal
to:

(
a)∑f
(x)

(
b)∑[
x+f
(x)
]

(
c)∑f
(x)
+x

(
d)∑xf
(x)
6.Whi
choft
hef
oll
owi
ngi
snotpossi
blei
npr
obabi
l
itydi
str
ibut
ion?

(
a)p(
x)≥0

(
b)∑p(
x)=1

(
c)∑xp(
x)=2

(
d)p(
x)=-
0.5
∞
7. ∫
-
∞
(
fx)
dxi
sal
way
sequal
to:

(
a)Zer
o

(
b)One

(
c)E(
X)

(
d)f
(x)+1

8.Av
ari
abl
ewhi
chcanassumef
ini
teorcount
abl
yinf
ini
tenumberofv
aluesi
sknownas:

(
a)Cont
inuous

(
b)Di
scr
ete

(
c)Qual
i
tat
ive

(
d)Noneoft
hem

9.I
fVar
(X)=5andVar
(Y)=10,
thenVar
(2X+Y)i
s:

(
a)15

(
b)20

(
c)10
 (
d)30

10.Anexpect
edv
alueofar
andom v
ari
abl
eisequal
toi
ts:

(
a)Var
iance

(
b)St
andar
ddev
iat
ion

(
c)Mean

(
d)Cov
ari
ance

11.Var
(4X+8)i
s:

(
a)12Var
(X)

(
b)4Var
(X)
+8

(
c)16Var
(X)

(
d)16Var
(X)
+8

12.Thedi
str
ibut
ionf
unct
ionF(
x)i
sequal
to:
(a)P(
X=x)

(
b)P(X≤x)

(
c)P(X≥x)

(
d)Al
loft
heabov
e

13.Thehei
ghtofper
sonsi
nacount
ryi
sar
andom v
ari
abl
eoft
het
ype:

(
a)Di
scr
eter
andom v
ari
abl
e

(
b)Cont
inuousr
andom v
ari
abl
e

(
c)Bot
h(a)and(
b)
 (
d)Nei
ther(
a)and(
b)

14.Theex
pect
ati
onoft
hesum oft
wor
andom v
ari
abl
esXandYi
sequal
to:

(
a)E(
X)E(
Y)

(
b)E(
X)+E(
Y)

(
c)E(
X±Y)

(
d)E(
XY)

15.Gi
venE(
X)=5andE(
Y)=-
2,t
henE(
X-Y)i
s--
--

16.I
fci
saconst
ant(
non-
random v
ari
abl
e),
thenE(
c)i
s--
--
-

17.Ali
sti
ngofall
theout
comesofanexper
imentandt
hepr
obabi
l
ityassoci
atedwi
theachout
comei
scal
l
ed:
(a)Pr
obabi
l
ity
di
str
ibut
ion

(
b)Pr
obabi
l
itydensi
tyf
unct
ion

(
c)At
tri
but
es

(
d)Di
str
ibut
ionf
unct
ion

18.Whatdoest
hesetcompr
isi
ngal
lpossi
bleout
comesofanexper
imentknownas?

a.Nul
lev
ent
 b.Sur
eevent
c.El
ementaryev
ent
d.Noneoftheabov
e

19.Mut
ual
l
yExcl
usi
veev
ent
s__
___
___

a.Cont
ainal
lsamplepoint
s
b.Cont
ainall
commonsampl epoints
c.Doesnotcont
ainanycommonsampl epoi
nt
d.Doesnotcont
ainanysamplepoint

20.I
nabox,ther
ear
e8r
ed,
7bl
ueand6gr
eenbal
l
s.Onebal
li
spi
ckedupr
andoml
y.Whati
sthepr
obabi
l
ityt
hati
tisnei
therr
ed
norgr
een?

21.I
ff(
x)=1/
10,
x=10,
thenE(
X)i
s:

(
a)Zer
o(b)6/
8(c)1(
d)-
1

22.Apr
obabi
l
itydensi
tyf
unct
ionber
epr
esent
edby
:

(
a)Tabl
e

(
b)Gr
aph

(
c)Mat
hemat
ical
equat
ion

(
d)Bot
h(b)and(
c)

23.Fi
ndVar
(X+8)
.

24.Adi
scr
etepr
obabi
l
ityf
unct
ionf
(x)i
sal
way
snon-
negat
iveandal
way
sli
esbet
ween:

(
a)0and∞

(
b)0and1
 (
c)-
1and+1

(
d)-
∞ and+∞

25.Gi
venE(
X)=10andE(
Y)=7,
thenE(
X-Y)i
s--
--

26.St
atewhet
hert
hef
oll
owi
ngi
sapr
obabi
l
itydi
str
ibut
ionsofar
andom v
ari
abl
e.Gi
ver
easonsf
ory
ouranswer
.

X 0 1 2

P(
X) 0.
4 0.
4 0.
2

27.St
atewhet
hert
hef
oll
owi
ngi
sapr
obabi
l
itydi
str
ibut
ionsofar
andom v
ari
abl
e.Gi
ver
easonsf
ory
ouranswer
.

X 0 1 2 3 4

P( 0. 0. 0. - 0.
X) 1 5 2 0. 3
1

28.St
atewhet
hert
hef
oll
owi
ngi
sapr
obabi
l
itydi
str
ibut
ionsofar
andom v
ari
abl
e.Gi
ver
easonsf
ory
ouranswer
.

X -
1 0 1

P(
X) 0.
6 0.
1 0.
2

29.St
atewhet
hert
hef
oll
owi
ngi
sapr
obabi
l
itydi
str
ibut
ionsofar
andom v
ari
abl
e.Gi
ver
easonsf
ory
ouranswer
.

X 3 2 1 0 -
1
 P( 0. 0. 0. 0. 0.
X) 3 2 4 1 0
5

30.Fi
ndE(
X)f
ort
hef
oll
owi
ng;

X 1 2 5

P(
X) 1 1 1
2 3 6

31.Whatar
ethepr
oper
ti
esofadi
str
ibut
ionf
unct
ion?

32.
Fi
ndki
ff(
x)= {
k,x=1,
0,ot
2,
3,
her
4,
wi
5,
6
se

{
33. 0.
2whenx=1
I
ff(
x)= 0.
3,whenx=2,
findE(
x).
0.
5whenx=3

34.Inameeti
ng,70%ofnumbersoft
hemember
sf avourand30%opposeacer
tai
npr
oposal
.Amemberi
ssel
ect
edatr
andom
andwetakeX=0,i
fheopposedandX=1i
fheisinfavour
.FindE(
X).

35. 1 1 1
Fi
ndE(
x)i
ff(
1)= ,f
(2)= andf
(3)= .
4 2 4

36.Whatar
ethepr
oper
ti
esofExpect
ati
on?

37.Whatar
ethepr
oper
ti
esofv
ari
ance?

38.Whatar
ethepr
oper
ti
esofpr
obabi
l
ityf
unct
ionofacont
inuousr
andom v
ari
abl
e?

39.Xi
sar
andom v
ari
abl
esucht
hatf
(x)
=2xf
or0˂
x˂1andf
(x)=0ot
her
wise.Fi
ndE(
x).
40.Def
inedi
str
ibut
ionf
unct
ionofacont
inuousr
andom v
ari
abl
e.

41.Inameeti
ng,70%ofnumbersoft
hemember
sf avourand30%opposeacer
tai
npr
oposal
.Amemberi
ssel
ect
edatr
andom
andwetakeX=0,i
fheopposedandX=1i
fheisinfavour
.FindE(
X).

42.Def
inej
ointpr
obabi
l
ityf
unct
ion.

43.Def
inemar
ginal
probabi
l
ityf
unct
ion

44.Def
inecondi
ti
onal
probabi
l
ityf
unct
ion

45.Whenar
etwov
ari
abl
esst
ati
sti
cal
l
yindependent
?

46.Showt
hatf
(x)=e-xf
orx>0i
sapr
obabi
l
itydensi
tyf
unct
ion.
4
47.I
ff(
x)=5(
1-x)and0≤x≤1.Fi
ndP(
x≥0.
5).
4
48.I
ff(
x)=5(
1-x)and0≤x≤1.Fi
ndP(
0.2≤x≤0.
8).

49.I
fxandyar
etwoi
ndependentv
ari
atesandV(
x)=2andV(
y)=3,
findV(
2x+3y
).

50.Ev
aluat
eki
fthef
oll
owi
ngi
sapr
obabi
l
itydensi
tyf
unct
ioni
s

X 0 1 2 3

P( 1 1 k 1
X) 6 2 10 30

51.Ev
aluat
eP(
1≤x≤3)i
fthef
oll
owi
ngi
sapr
obabi
l
itydensi
tyf
unct
ioni
s

X 0 1 2 3

P( 1 1 3 1
X) 6 2 10 30
52. 1121 1
Ar
andom v
ari
abl
ext
akesv
alues0,
1,
2,
3,
4wi
thcor
respondi
ngpr
obabi
l
iti
es ,,,, .Fi
ndt
heexpect
ati
onofX.
455840

53.Xi
sar
andom v
ari
abl
ewhosemeani
sμandst
andar
ddev
iat
ionσ.Whatwi
l
lbet
hemeanandst
andar
ddev
iat
ionof2x.

54.Xi
sar
andom v
ari
abl
ewhosemeani
sμandst
andar
ddev
iat
ionσ.Whatwi
l
lbet
hemeanandst
andar
ddev
iat
ionof2x+1.

55.Whatar
ethepr
oper
ti
esofj
ointpr
obabi
l
ityf
unct
ion?

56.Ev
aluat
eki
fthef
oll
owi
ngi
sapr
obabi
l
itydensi
tyf
unct
ion.Al
soobt
ainP(
1≤x≤3)
.

X 0 1 2 3

P( 1 1 k 1
X) 6 2 10 30

Ar
andom v
ari
abl
exf
oll
owsapr
obabi
l
itydi
str
ibut
ionasgi
venbel
ow.

Fi
ndmeanandv
ari
anceoft
hev
ari
abl
e.

X 0 1 2 3

P(
X) 5 5 14 1
18 27 27 54
 1 2 3
Ar
andom v
ari
abl
eXhast
hef
oll
owi
ngpr
obabi
l
ityf
unct
ionf
(0)= ,f
(1)
= , f
(2)= andf
(x)=0ot
her
wise
6 6 6

i
) Wr
it
edownt
hedensi
tyf
unct
ionanddi
str
ibut
ionf
unct
ion.

i
i
) Fi
ndP(
x<2)

i
i
i) Fi
ndP(
0<x<2)

LetXdenot
ethenumberofheadsobt
ainedi
nthreesuccessi
vet
ossesofacoin,t
hepr
obabi
l
ityoff
all
i
ngheadsf
orwhi
chi
na
2
si
nglet
ossis .Fi
ndthepr
obabil
it
ydi
stri
but
ionofXandcomputeitsexpect
ati
on.
3

Thr
eebasket
scont
ainsr
espect
ivel
y3gr
eenand2whi
tebal
l
s,5gr
eenand6whi
tebal
l
sand2gr
eenand4whi
tebal
l
s.

Onebal
li
sdr
awnf
rom eachbasket
.Fi
ndt
heexpect
ednumberofwhi
tebal
l
sdr
awnout
.

Xi
sar
andom v
ari
abl
ewhosemeani
sμ

andst
andar
ddev
iat
ionσ.Whatwi
l
lbet
hemeanandv
ari
anceof:

1)4X

2)3X+4

Xi
sar
andom v
ari
abl
ewhosemeani
sμ

andst
andar
ddev
iat
ionσ.Whatwi
l
lbet
hemeanandv
ari
anceof

1)3X

2)12X+40
Xi
sar
andom v
ari
abl
ewhosemeani
sμ

andst
andar
ddev
iat
ionσ.Whatwi
l
lbet
hemeanandv
ari
anceof

1)8X

2)3X+74

Fi
ndt
hemeanandv
aranceofax+bandx2,
i if

̅
X =4andV(
X)=8.

Def
ine,
fordi
scr
eter
andom v
ari
abl
esXandY

1)Joi
ntpr
obabi
l
itymassf
unct
ion

2)Mar
ginal
probabi
l
itymassf
unct
ion

3)Condi
ti
onal
probabi
l
itymassf
unct
ion

Defi
neprobabil
i
tydensi
tyfunct
ionanddi
str
ibut
ionf
unct
ionofacont
inuousr
andom v
ari
abl
edef
inedi
nthei
nter
val
(a,
b),
andli
stoutthei
rproper
ti
es.

I
ff(
x)=k(
10-
x)x2i
sapr
obabi
l
itydensi
tyf
unct
ionf
or0≤x≤1.Fi
ndk?

I
ff(
x)=5(
1-x)4,
0≤x≤1.Fi
nd

1)P(
X≥0.
5)

2)P(0.
2≤x≤0.
8)
Av
ari
abl
eXhast
hepr
obabi
l
itydensi
tyf
unct
ionf
(x)=6x(
1-x)
,0≤x≤1.Fi
ndmeanofX.

Fi
ndt
hef
ir
stt
womoment
saboutmeanofar
andom v
ari
abl
eXwhose

f
(x)
=2(
1-x)
,0≤x≤1.

Xi
sar
andom v
ari
abl
ewhosemeani
sμ

.Whatwi
l
lbet
hemeanof
:

i
) 2

i
i
) 5x

i
i
i) 7x+3

i
v) 2x2

Xi
sar
andom v
ari
abl
ewhosest
andar
ddev
iat
ionσ.Whatwi
l
lbet
hev
ari
anceof
:

i
) 1

i
i
) 4x

i
i
i) 6x+8

i
v) 5x-
3

LetXdenotesthenumberofheadsobtainedinthr
eesuccessi
vet
ossesofacoin,thepr
obabi
l
ityoff
all
i
ngheadsf
orwhi
chi
n
2
asingl
etossis .Findt
heprobabi
li
tydistr
ibut
ionofXandcomputeit
sexpect
ation.
3
Xi
sar
andom v
ari
abl
ewhosemeani
sμ

.Whatwi
l
lbet
hemeanof
:

i
) 7

i
i
) 4x

i
i
i) x+3

i
v) x3

Xi
sar
andom v
ari
abl
ewhosest
andar
ddev
iat
ionσ.Whatwi
l
lbet
hev
ari
anceof
:

i
) 100

i
i
) 10x

i
i
i) 9x+4

i
v) x-
3

Fi he3rdand4th moment
ndt saboutmeanofar
andom v
ari
abl
eXwhose

f
(x)
=2(
1-x)
,0≤x≤1.

Ar
andom v
ari
abl
eXhast
hef
oll
owi
ngpr
obabi
l
itydi
str
ibut
ion:

X 0 1 2 3 4 5 6 7

P( 0 k 2 2 3 k2 2 7
X) k k k k2 k2
+k
Det
ermi
ne;

i
) k

i
i
) P(
X<3)

Fi
ndt
hemeannumberofheadsi
nthr
eet
ossesofaf
aircoi
n.

Twodi
cear
ethr
ownsi
mul
taneousl
y.I
fXdenot
est
henumberofsi
xes,
findt
heexpect
ati
onofX.

LetXdenot
est
hesum oft
henumber
sobt
ainedwhent
wof
airdi
cesar
erol
l
ed.Fi
ndt
hemeanofX.

a)Defi
nearandom v
ari
able.Whatdoyoumeanby(
i)di
str
ibut
ionf
unct
ionofacont
inuousr
andom v
ari
abl
e(i
i
)
expect
ati
onofacont
inuousrandom v
ari
abl
e?

b)Ev
aluat
ek,
ift
hef
oll
owi
ngi
sapr
obabi
l
itydi
str
ibut
ion.Al
sof
indP(
x≤2)andP(
0˂x<3)
.

k k k k
f
(0)= ,f
(1)= ,f
(2)= , f
(3)= ,andf
(x)=0el
sewher
e.
2 5 20 4
-
m x
em
c) Fi
ndt
heMomentgener
ati
ngf
unct
ionofXwhosef
(x)= f
orx=0,
1,
2,
..
.
x!

a)Ther
andom v
ari
abl
eXhast
hepr
obabi
l
itydi
str
ibut
ionP(
X)oft
hef
oll
owi
ngf
orm;
 {
k,
ifx=0
2k,i
fx=1
P(
X)=
3k,i
fx=2
0,ot
herwi
se

i
) Det
ermi
net
hev
alueofk

i
i
) Fi
ndP(
X<2)
,P(
X≤2)
,andP(
X≥2)

b)Twonumbersar
eselectedatrandom wi
thoutr
epl
acementf
rom t
hef
ir
st6posi
ti
vei
nteger
s.LetXdenot
ethel
arger
oft
hetwonumbersobtained.Fi
ndE(X).

a)Fi
ndt
hepr
obabi
l
itydi
str
ibut
ionofnumberoft
ail
sint
hesi
mul
taneoust
ossesoft
hreecoi
ns.

b)LetXdenot
ethesum oft
henumber
sobt
ainedwhent
wof
airdi
esar
erol
l
ed.Fi
ndt
hemeanv
ari
anceandst
andar
d
devi
ati
onofX.

a)Aclassof15st udent
swhoseagesare14,
17,
15,
14,21,
17,
19,
20,16,
18,
20,
17,
16,19and20y ears.Onestudentissel
ect
edin
suchmannerthateachhasthesamechanceofbei
ngchosenandt heageXoftheselectedst
udentisrecorded.Whatist
he
probabi
li
tydi
str
ibuti
onofther
andom var
iabl
eX?Findmean,var
ianceandst
andarddev i
ati
onofX.

a)Fi
ndt
hef
ir
stf
ourmoment
saboutt
hemeanoft
her
andom v
ari
abl
e,Xwhosef
(x)=4(
1-x)f
or0≤x≤1.

b)Fi
ndt
hedi
str
ibut
ionf
unct
ionofXwhosef
(x)=2e-2xf
or0<x<∞.
4
c) I
ff(
x)=5(
1-x)and0≤x≤1f
ind
P(
x≥0.
5)

a)
Twocar dsar
edr
awnsuccessivel
y(wi
thr
epl
acement
)fr
om awel
lshuf
fl
eddeckof52car
ds.Fi
ndt
hepr
obabi
l
ity
di
str
ibut
ionoft
henumberofaces.

b)Ev
aluat
eki
fthef
oll
owi
ngf
unct
ioni
sapr
obabi
l
itydi
str
ibut
ion.
 {
k
i
fx=0
2
k
ifx=1
5
f
(x)= k
i
fx=2
20
k
ifx=3
4
0otherwi
se

i
) Ev
aluat
ek.

i
i
) P(
X≤3)

i
i
i) P(
X<2)

i
v) E(
X)

v
) V(
X)