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STAT 6201 - Mathematical Statistics
A box contains three coins with a head on each side, four coins with a
tail on each side, and two fair coins. If one of these nine coins is selected
at random and tossed once, what is the probability that a head will be
obtained?
the
Define
T
,
selected
=
IT
=
selected
T3= selected
A
=
H
will
events
following
win
coin
be
is
is
Prlt D=
H only
is
win
T
Prltz)= ht
only
fair
(
H
3T
-
T
)
MB )=±a
obtained
PrlA)=Pr( TDPRCAITDTHHDPRCAITDTPRCTDRCAITD
Pr ( Alt
,
)
=
PRCAITD
=
IHAITD
=
1
0
I
Consider an experiment in which a fair coin is tossed until a head is
obtained for the first time. If this experiment is performed three times,
what is the probability that exactly the same number of tosses will be
required for each of the three performances?
Pr
(
same
#
)
tosses
of
=
Pr ( each
+
Pr ( each
+
Bleach
Pr
( each
takes
exp
exp takes
=a3+# 't
.
.
n
toss
1
.
.
)
.
'
exp
.
.
I
't ) (
takes
takes
exp
tz
=
)= ( E
tosses
.+KTt
Pr ( each
lakes
experiment
.
2
exaoty
tosses
tosses
3
)
.
's
toss
)
+
=
.
.
t.es
I ? 'z)( It
I
) =Hpn
.
)
Suppose that X , Y have a continuous joint distribution for which the
joint pdf is
(
cy 2 if 0  x  2 0  y  1
f (x, y ) =
0
otherwise
1. Determine the value of the constant c.
2. Find Pr(X + Y > 2).
1
.
2
.
fj
'
Spy dxdy
Prc xty
>
2)
=
=
{
Sj cy ?
2
dg
htycgidxdy
2c
=
=
.
9531.1
fkj
(2
=
-
2 c.
Hy )
's
dy
=
=L
c.
Ce
'
Hgl
=
.
I
÷
Suppose that the joint pdf of two random variables X and Y is as follows
(
3
(4 2x y ) if x > 0, y > 0, 2x + y < 4
f (x, y ) = 16
0
otherwise
1. Determine the conditional pdf of Y given X .
2. Find Pr(Y
2 | X = 0.5).
"
gyylxj.tk#filH=Fafkihdy=f4j2
f 't 'd
=
}u( 4-2,4
-
( 4-2×1
-
the
}u(4rx-y)dy
.tl
4-2×12=76.1214-2×5
=hf(
gdylx )=To(
O<x<
4-2×7 )
.
Is
.¥
,
=
.
2-
4
xp
'T
2( 2+42
2
o<x<
o<
y
2
<
4-
Zx
Pr ( y >
,
2
I
X
=
0.5 )
[
=
=
2.
Th ( y
I
ftp.t.ay
.
5) dy
=
exercise
Suppose that either of two instruments might be used for making a
certain measurement. Instrument 1 yields a measurement whose pdf h1 is
(
2x if 0 < x < 1
h1 (x) =
0
otherwise
Instrument 2 yields a measurement whose pdf is h2 is
(
3x 2 if 0 < x < 2
h2 (x) =
0
otherwise
Suppose that one of the two instruments is chosen at random and a
measurement X is made with it.
1. Determine the marginal pdf of X
2. If the value of the measurement is X = 1/2 what is the probability
that instrument 1 was used.
Let
be
Y
the
variable
random
which
indicating
instrument
being
is
used
yell ,2}
The
X
the pdf of
Since
|Y=1
X
of
pdf
/
Y=2
instrument
aw
h
is
is
is
,
(
x
)
hdx)
selected
at
Pr(Y=D=Yz
random
( this defines
just
The
FKIY )
pmf
/ pdf
of ( X ,Y )
y±2
y=l
the
=
{
Prly =D
.
h.cn )
pr(y=y hdx )
.
marginal
-
pdf of
and
the
Pr(y=2)=Yz
marginal
is
if
if
X
is
YEFHIY
)
=
=
PrlY=Dh,(
th
,
(a)
+
H
{
+
Pr(Y=z)hdx)=
hdx )
pmf of Y )
conditional
the
gdylx
th
"
)
pmf
ffky
#
of
)
Y
given X
is
joint
marginal
( n)
'
g
,aH=prly=i1x= =¥Y=±"h
gdzlx )=Pr(y=z1x d=fty÷=
Pr(Y=l1X=k)=gd
1
los )=
±hl0
.
tzhk
th ,( Ntt
"
tzhdashtzhdo
's )
'
hdx )
h+Y÷
,
In a large collection of coins, the probability X that a head will be
obtained when a coin is tossed varies from one coin to another, and the
distribution of X in the collection is specified by the following pdf:
(
6x(1 x) if 0 < x < 1
f (x) =
0
otherwise
Suppose that a coin is selected at random from the collection and tossed
once, and that a head is obtained. Determine the conditional pdf of X
for this coin.
Let
C
Say
Ctb
be
,
I
the
}
indicates
C =O
The
information given
Pr(C=1
/
X=x
)
Pr(C=o|X=x)=
=
x
that
that
coin
outcome
.
is
Outcome
that
the
x
1-
is
selected
indicates
C 't
where
the
of
outcome
marginal
fk
)
pdf of
X
is
H
is
T
the joint
f(
the
x
,y)=µH
flx )
problem
of
pmf/pdf
is
it
"
.
(
tx
)
asking
X,
Y
C
is
"
( note
if y=o
for
the
how
conditional
pdf
Thus
marginal
of
the
write
X
joint
gdxl C=D=%(x1y=D=
the
we
joint
C=l
given
marginal
=
,
that
conditional
x
is
f#=D
,
=
3(
marginal
C
Pr(C=D=§xfHdx=§6x4ix)dn =L
the conditional pdf of X given
that
C
't
is
t.pt#=zxf(x)=12x4rx
)
,
o<x<
i
)
Suppose that random variables X1 , X2 , X3 are iid with a discrete
distribution for which the pdf is f . Determine the value of
Pr(X1 = X2 = X3 )
=
Say
independent
X,
and
,
so
id identically
and
,dz
Ek
,=x
,
.
.
}
.
)=Pr(×z=d=Pr(X5a
Pr(
X
÷X
Prlx
distributed
,
,
)=f(
,=dn)=Pr(Xe4=Pr(Xs=4=
,=X<=XD=FgH×i×z=×s
-
Ln
)
-
xD
fkn )
fkitt FKDI
.
.
.+fW 't
...
Suppose that the n random variables X1 , . . . , Xn form a random sample
from a continuous distribution for which the pdf is f . Determine the
probability that at least k of these n random variables will lie in a
specified interval [a, b].
The
probability
desired
(nu)
between
is
p
a
'
Ku ) put 'apT"
pkapshht
where
is
the
and
+
.
probability
b
p= Pr(
a
<
X,
Eb
)
'
=
.
.tk ) Fapi
that
[
one
floddx
random
variable
lies
.
Suppose that a random variable X can have each of the seven values
3, 2, 1, 0, 1, 2, 3 with equal probability. Determine the pmf of
Y = X2 X.
-
-
The
possible
for
values
Pr(Y=o)=Pr(X=
Pr(Y=2)=Pr(X=
1
-1
}
Pr(y=6)=Pr(X=
My -tz)=Pr(x=
or
or
Y
are
X=o
)
11=2 )
2
-
-
3)
or
=
=
0,2
=
}
X=3)=÷
's
,
6
12
,
Suppose that the pdf of a random variable X is
(
x
if 0 < x < 2
f (x) = 2
0 otherwise
1. Determine the cdf and pdf of Y = X (2
2. Determine the pdf of Y = 3X + 2.
3. Determine the pdf of Y = 4 X 3 .
with 2
start
First
find
{
X
"
.
the
cdf
of
X
[nfHdz
FC a )=
If
X ).
<
0
Flx )=0
=⇐E=÷
fokndz
Hatted
extort
new
;t;
-
find
Now
off
of
the
)=Pr(Y±y)=Pr(
Gly
first
Y
3×+2
<
g)
Xefly
Pr(
-
the
using
,
-
D)
F
=
( fly )
.
FC )
2)
is
.
the
cdf
of X
¥dotX
>:(
fly
If
2)
0±
G(y)=o
0
<
2
ya
or
If
-
fly
-4<2
or
y78
or
-52=1%4.25
Gly )=l
fly -432
If
'=H( th
z£y<
64
8
c¥
)=€b(
0
of
Y
Gly
is
y
.
zp
1
Now
the
pdf
=
of
Y
is
fly )=
if
y< 2
if
2ey<
if
8
y
g)
-
dy
fly )=O
if
=tz ( Zy -4 )=
y<
2
or
y
holy 2)
-
>
8
if
z±y<
8
3
.
Find
the
Note
that
H(z)= Prl
Jf
0
depends
Z
e =Pr(
Pr (
z
,
8
X3
find
,
the
cdf
of
2=1/3
X3£
0
=
H(z)=0
<
z
×
c
z
Pr (
X3Ez)=
's
=Pr(
Pr(X3<
z
±
Xez
"'
)=F(z' 3)
's
'
=
HE ) ?
tyz43
I
it
Hc ={YaE
Into
'
so
if
1
Now
X3
-
on
X3<
8×3
_<
<
It
4=4
Y
z< 0
If
of
pdf
,
find
the
cdf
Gly )=Pr(Y±y)=Pr( 4-
zz
8
X? 4
of
4=4
Z
±y)=Pr(
-
z
,
-
2
4- y )= 1-
HC 4- y )
cdf
Rest
of
the problem
is
left
as
aw
exercise
HE )
where
of
Z
is
the
Suppose that a random variable has the uniform distribution on the
interval [0, 1].
t
/
2
1. Determine the pdf of X
2. Determine the pdf of X 3
tint
find
the
lheu
find
the
find
the
cdf
of
X
first
call
it
FED
exercise
:
then
Repeat
,
,
for
Y=
Cdt
-
1/3
Y=X2
of
pdf of
,
Y
Suppose that the pdf of X is
f (x) =
(
x
e
0
if x > 0
otherwise
Determine the cdf and pdf of Y = X 1/2
Find
the
cdf
of
X
Flx )=Pr(×±x )
if
if
50
o
F(x)=o
x%o
FCx)=
x<
Ffx )
-
{
it
°
1-
"eYdy
So
It
if
no
an
=
-
ist 's
=
1
-
et
Now
find
the
cdf
Gly )=Pr( Yey )
If
y
<
Gly )
=
"fp
↳
T
=/
.
The
pdf
xkey)
Pr(
=
Y
Gly )=o
o
y70
GH
of
g
of
Y
Pr
( xkey ) =p (
,
( take
.
is
of
°
hw=ddfYT={
the calf
so
"
Zyet
if
ya
'
×
c-
of
y
)
X
=
Fly
'
)
F
where
is
the cat
'
and
plug
-
in
y
in
place
of
of
a)
X