stat 6201 - mathematical statistics · suppose that x,y have a continuous joint distribution for...

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?

Define the following events

T,

= selected win is H onlyPrlt D= 3T

IT = selected win is T onlyPrltz)= ht

T3= selected coin is fair ( H - T ) MB )=±a

A = H will be obtained

PrlA)=Pr( TDPRCAITDTHHDPRCAITDTPRCTDRCAITD

Pr ( Alt , ) = 1

PRCAITD = 0

IHAITD = 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 # of tosses ) = Pr ( each experiment lakes exaoty I toss)+ Pr ( each exp .

takes 2 tosses )+ Pr ( each exp takes 3 tosses ) +

.. .

=

Bleach exp takes 1 toss ) = tz . I . t.esPr (each exp takes n tosses )= ( E

'

't ) ( I ? 'z)( It . 's) =Hpn

=a3+# 't . . .+KTt . . .

Suppose that X ,Y have a continuous joint distribution for which the

joint pdf is

f (x , y) =

(cy

2if 0 x 2 0 y 1

0 otherwise

1. Determine the value of the constant c .

2. Find Pr(X + Y > 2).

1. fj Spy

'

dxdy = Sj cy ? 2dg = 2c . 9531.1 =

2 c. 's =L Ce I

2 .Prc xty > 2) = { htycgidxdy = fkj (2 - Hy ) dy = c. Hgl

.

'= ÷

Suppose that the joint pdf of two random variables X and Y is as follows

f (x , y) =

(316 (4� 2x � y) if x > 0, y > 0, 2x + y < 4

0 otherwise

1. Determine the conditional pdf of Y given X .

2. Find Pr(Y � 2 |X = 0.5).

gyylxj.tk#filH=Fafkihdy=f4j2"

}u(4rx-y)dyf 't 'd

= }u(4-2,4 - ( 4-2×1 - the .tl4-2×12=76.1214-2×5

=hf( 2- xp

O<x<2

gdylx )=To( 4-2×7 ) . Is .¥, .=

4 'T o<x< 2

2(2+42 o< y < 4- Zx

Pr ( y >, 2 I X = 0.5 ) = [2.

Th (y I. 5) dy

= ftp.t.ay = exercise

Suppose that either of two instruments might be used for making a

certain measurement. Instrument 1 yields a measurement whose pdf h1 is

h1(x) =

(2x if 0 < x < 1

0 otherwise

Instrument 2 yields a measurement whose pdf is h2 is

h2(x) =

(3x

2if 0 < x < 2

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 Y be the random variable indicating which instrument is being used

⇒ yell ,2}

The pdf of X |Y=1 is h,( x )

the pdf of X / Y=2 is hdx)

Since aw instrument is selected at random ⇒ Pr(Y=D=Yz and Pr(y=2)=Yz

(this defines the marginal pmf of Y )

The justpmf / pdf of ( X ,Y ) is

Prly =D . h.cn) if

y=lFKIY ) = { pr(y=y . hdx) ify±2the marginal pdf of X is YEFHIY ) = PrlY=Dh,( H + Pr(Y=z)hdx)=

-

= th ,(a) + { hdx )

the conditional pmf of Y given X is

ffky ) joint

gdylx ) #( n ) marginal

th "'

g ,aH=prly=i1x=⇒=¥Y=±"h h+Y÷ ,

gdzlx )=Pr(y=z1x⇒d=fty÷=tzhk '

th ,( Ntt hdx )

Pr(Y=l1X=k)=gd 1 los )=±hl0 . "

tzhdashtzhdo 's )

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:

f (x) =

(6x(1� x) if 0 < x < 1

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 be the outcome of the selected coin .

Say Ctb ,I } where C 't indicates that outcome is H

C =O indicates that Outcome is T

The information given is that

Pr(C=1 / X=x ) = xthe marginal pdf of X is

Pr(C=o|X=x)= 1- xfk )

the joint pmf/pdf of X,

C is

f( x ,y)=µH. " it Y "

( note how we write the joint = marginal x conditional )

flx ) ( tx ) if y=o

the problem is asking for the conditional pdf of X given C=l,

that is

gdxl C=D=%(x1y=D= joint ,

marginal= f#3( C =D

the marginal Pr(C=D=§xfHdx=§6x4ix)dn =L

Thus 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)

id= independent and identically distributed

Say X, Ek, ,dz,

. . . }

and so Pr( X,=x,

)=Pr(×z=d=Pr(X5a, )=f( xD

÷ X ,=dn)=Pr(Xe4=Pr(Xs=4= fkn )

Prlx ,=X<=XD=FgH×i×z=×s - Ln ) - 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 desired probability is

(nu) pkapshht Ku) put 'apT"' '

+. ..tk ) Fapi

where p is the probability that one random variable lies

between a and b

p= Pr( a < X,

Eb ) = [ floddx

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 = X

2 � X .

. - -

The possible values for Y are 0,2 ,6

,12

Pr(Y=o)=Pr(X= 1

or X=o ) =

}Pr(Y=2)=Pr(X= -1 or 11=2) = }

Pr(y=6)=Pr(X= - 2 or X=3)=÷

My -tz)=Pr(x= -3) = 's

Suppose that the pdf of a random variable X is

f (x) =

(x

2 if 0 < x < 2

0 otherwise

1. Determine the cdf and pdf of Y = X (2� X ).

2. Determine the pdf of Y = 3X + 2.

3. Determine the pdf of Y = 4� X

3.

start with 2.

First find the cdf of X

FC a )= [nfHdz

If X < 0 Flx )=0{ " extortnew;t;Hatted- fokndz =⇐E=÷

Now find the

offof Y first , using the

¥dotXGly )=Pr(Y±y)=Pr( 3×+2 < g) - Pr( Xefly - D) = F ( fly . 2)) FC . ) is the cdf of X

If fly - 2) < 0 ⇒ G(y)=o

or ya 2

If 0± fly -4<2

orz£y<8

64 '=H( th -52=1%4.25

If fly -432 Gly )=l

or y780 if y< 2

c¥of Y is Gly )=€b(y . zp if 2ey< 8

1 if y

>:(g)

Now the pdf of Y is fly )= - =tz (Zy -4 )= holy - 2) if z±y< 8

= dy

fly )=O if y< 2 or y > 8

3.

Find the pdf of 4=4 - X3

Note that Y depends on X3, find the cdf of 2=1/3

H(z)= Prl Z e⇒=Pr( X3£ ⇒Jf z< 0 ⇒ Pr ( X3< ⇒ = 0 ⇒ H(z)=0

If 0 _< z <8×3< z c ⇒ × ± z's

⇒ Pr(X3< ⇒=Pr( Xez" ' )=F(z' '3) = HE's ) ? tyz43

It z , 8Pr ( X3Ez)= I

so Hc⇒={YaE' itInto1 if zz 8

Now, find the cdf of 4=4 - X? 4 - 2

Gly )=Pr(Y±y)=Pr( 4- Z ±y)=Pr( z , 4- y )= 1- HC 4- y ) where HE ) is the

cdf of Z

Rest of the problem is left as aw exercise

Suppose that a random variable has the uniform distribution on the

interval [0, 1].

1. Determine the pdf of X

2

2. Determine the pdf of �X

3

t/

exercisetint : find the cdf of X first , call it FED

lheu, find the Cdt of Y=X2

then, find the pdf of Y

Repeat for Y= -1/3

Suppose that the pdf of X is

f (x) =

(e

�x

if x > 0

0 otherwise

Determine the cdf and pdf of Y = X

1/2

Find the cdf of X

Flx )=Pr(×±x )

if x< o F(x)=o

if x%o FCx)= So"eYdy = - ist 's = 1 - et

50 Ffx ) - {° it no

1- It if an

Now find the cdf of Y

Gly)=Pr( Yey ) = Pr( xkey)If y

< o ⇒ Gly )=o

y70 ⇒ Gly ) = Pr ( xkey ) =p , ( × c- y' ) = Fly ') where F is the cat of X

GH =/. g"f↳Tp .( take the calf of X and plug - in y

'

in place of a)

The pdf of Y is

hw=ddfYT={° of so

Zyet"

if ya