ECE 250 Midterm October 28, 2016

SOLUTION

1. The discrete random variable X has the probabilities P(X = -2) = P(X = -1) = P(X = 1) = P(X = 2) = 1/4. The random variable Y is defined by Y = independent?

lxJ.

Are X andY uncorrelated? Are they

I Krkj-t-..,A,.;f;- tff- P(~.: n1y;: wr) =P~::. Yt) P( '(:= wr)

fOr ALL.. n) WI

An answer not supported by appropriate reasoning will not receive any credit.

-2 -\

I :J-..

Y= lZI XY- Prob. 2 -. 4'I+ I

-.1

I 2

l

E. fXiJ

t

1/4 1

/4

'/+

: : (-4)xP(X<"72)~(-')~
£LX) :: (- 2) P(X= -2.) +( -1) P(X=-1)+-( l)f[X=t) -f(2)P(K= 2 ) :.0

ELYj : \2) P< X= - 2)

=t

_·.

+(I) P(X=-1) 4-(1 )P(X;:I )+2 P(X= ;z)

ELXYJ = ~[2CJELY1

P( X= -;z., Y~ 2) "' ;f- j P(X.,-z)= ;( j P[ Y: 2) PC Z =- 2) P( '1:: 2) =- -ft;- =t PC X"= - 2 7 Y= z) 5vffcer..J~t- To ()&;It a. .. y ffJ•rdt vo. rc G\ \Jt.es X a"o{ Y

..

No

Uncorrelated? Independent? (Circle One)

=f

Yes

Page 2 of9

-· '

'

SOLUTION 2. The random variables X and Y have the joint characteristic function

1

1

2

2

xy(u, v) = -cos(u- v) + -cos(u + v).

'

Are X and Y independent?

An answer not supported by appropriate reasoning will not receive any credit.

CE x: (l<) ~_g(.v) =-

Co '7U. w~\1

= {f.I',.f (u1v)

= ±~ v.-v) + ~COS{?IN) ,..---.fy. o wr ''Us.t{vl" for..,Ja~ll

Independent? (Circle One)

No

Page 4 of 10

•

SOLUTION

3. The random variables X and Y are independent and identically distributed with probabilities 1 n 1 m P(X=n)=(-) and P(Y=m)= (-) , n,m= 1,2, ....

2

2

Evaluate the probabilityP(X ~ Y).

= ~~

P(X=

Vt)

~ == vn) ~

V\.Zwt

00

=

2:

V\

f(X~~) ~

Vl=-l

I

P(Y:: WI)

,..:J

"

)l.

JIYI

:£ P(Y= v.,}:: 2_(jj .....,::,

Wt:=l

::. 7Jl- (-5)V1

I

P(X ;>: Y)

~ Page 6 of 10

---

--------

~-----

-----

-

SOLUTION 4. The random variables Xand Y have the joint density

fx,Y (x,y) =

2, O~x~y~l { O h . , ot erwzse.

Determine the probability density of Z = X + Y. You must provide values of the density for all possible values of z.

An answer not supported by appropriate reasoning will not receive any credit.

}}-&,'l~J,)dtcdy

xfr~r

..

Page 8 of9

i

'

ECE 250 Final Exam December 8, 2016

SOLUTION

1. Consider the random variables

Here N 1 and N2 are independent Poisson variables

Uncorrelated? Independent? (Circle One)

8 Yes

Page 2 of 17

No

SOLUTION

2. The random variable X has a continuous probability density. The median of X is the value M that satisfies

Fx (M) = 1- Fx (M). Let W be the value that minimizes E[IX- WIJ. Prove that W = M.

Page 4 of 17

SOLUTION

3. N(t) is a Poisson process (independent increments, N(O) = 0, constant rate A.). Show that, for c > 0,

:i: ~ N;t)- AI~ e)~ o. 00

C h..e by'> [4ev

IVl~oa(t t

f(/ ~ - >.) ~,) ~

Va r ( ~- ;)

[HINT: Begin by evaluating the moments ofN(t).]

....

,

'

Page 6 of 17

----~-----~---'

SOLUTION 4. The non-random function g(t) is periodic with period T. That is g(t + T) = g(t). The random variable A is uniformly distributed on the interval [0, T]

fA(a)=

2_ { r' 0

O~a~T otherwise.

'

Consider the random process X(t) = g(t +A). Is X(t) wide sense stationary?

An answer not supported by appropriate reasoning will not receive any credit.

E[ XW] =- f"~(t-1-~'<){.',(,._)o(~ ;; 5~a.('t+ot)ol~ -~

~ Jv.te~val of q

~~;~og~ ;-;~':; LS

0

=

T 0

9.et g

t+T

r

= t fo(

t Llj(~>4(3 =

COM(jf;:

COMSi-

"

: . t 1)( tt-'i" >t< HA ~)cl"'­ = t- nt(tt-1:" ¥) ?ft+...)dol. 0 +oC

i;-14

-+f ~('t+(3)~(

5~ t ~::. t"+-d..

t+T

V

~=.

(J) d (3 .

~('C+~)~LP) r~

T

V

tf ?('C+~)~{f')o{p 0

= tarrcf,o.., W.S.S.?

No

(Circle One)

Page 8 of 17 I I________

f~r,odtc

fo., ction w1tn f~f'lo.l

CJ~ I y of

•

0\

77

T

SOLUTION 5. Consider the Poisson process N(t) (independent increments, N(O) = 0, constant rate A.). Denote by ~k the time between the (k- 1)st event and the k + n event. Let TM be the time until the M-th event

Th-e. f, »J~?

b~w-U.~? ~Jj Ct l.Qytt- ~V-tt~tfS lt ~ l Vl~p4Hcl4111t

M

TM=

IAt

k=l

Evaluate the mean and variance ofTM.

If

.ev-e,.,t5 are

t ..,cJ.ep- J r}te V a i't a VI r.l o1f o. ~ u VII l?

TJ..tt SOYI1

o { tk' llarz QlfCQ. ~

IE-[-TM_]_=_M_>_ _ _ _ _ _ _v_ar_[_TM_]_=_~_z._ _ _ _ _ _ L-

Page 10 of 17

___J\

SOLUTION 6. A zero-mean, white Gaussian process with constant power spectral density Po is passed through a linear, time-invariant filter

A ltnear

Y(t) = s:h(a)X(t-a)da. Here

h(t) =

sinm

--,

-00

< t < 00,

. Jrt

Evaluate the probability density ofY(t).

op.eratron OIA a ft"O uss r~sol'ts

GauGt;ra~ 1 n ct111o tker &aus~ta~ frrJCilSS . .

. . . Y (1)

tS,

a G-ao"'- var1 ct~le

An answer not supported by appropriate reasoning will not receive any credit.

+:rc·o C)J

t~-wr) :z

=

m: ~LYtt1] Va r[Yt-ln

J.r e;-::z.r:z.

___,___-==-= fvYJ ~ t-ra H s-1-or I ('..,___..,..-_ _

Hfiw)= r-Bct(~ )

=Po rtct( *1

K'

z

;:;

-::. { Po ) -'IT ~Wf. rr ')X ( W) 0 ) otJ..QrWI'-e, (r.-) ... J.,. l rr ;w~ ~ "211" ) d.w ~ Po ~/)1JTr

Pee

-

-'tT

l ) -n-~w~ n{ 0) CJTittlrwr s..e

=~

rr'Y;

~

:: Shlrf) X{t-o<)ol~ = o

E2 [ Yt-t)J

~

..

I.___ fv-(t)(-x)_=

...

~---~-~_:_ _ _ _ _ _ _ _---'1

_V_:Z_Tr_Po__

Page 12 of17

•

SOLUTION 7. N(t) is a Poisson process (independent increments, N(O) = 0, constant rate A).

P(N(t)- N(s) = n) = [A-(t ~s)f e-A,(t-s), n.

n =0,1, .... t ~

s.

Evaluate the conditional expectation E[N(t+r)IN(t)=m],

r~O,

m~O.

An answer not supported by appropriate reasoning will not receive any credit.

P( N('t~)=.., l N lt)=~ )::! P({Nl-l--+'L)-NHJJ-t-Nit)=- n fi.J lt)= Yn) N Lt-+'f )- Nl t)1 aKo/lJ lt}

a.~ 1Kd..e.~..fl.va ~tti.OK- ov-erlo.ppo .. r ~vt.T'.ervals

:. F({N(t+'i)-NLi--J}::Vl-n,) ~lt)=~) ::: p( {N (t+~)- N [-1:)~=- n- n, N ( t)=- ~)

-

? ::

Now E..( N l+-t't) l N [t):: V)'J]

e

=.

)

P( ~It) =-m)

P( ~(++'C)-IJ{t)= VJ-m) ~ p ~

zn

~~m~

( N l '1:-fJL) =V7 l N [i):::. m)

~

«»

:3>

2:.., P(NltH)-1./It> .. VI-In) ~ =="'~(k +-~"') f'( Nl-t¥f)- Nlt)::. P.) =

()()

(~

) ...

:: 2:_( /cz+m) 01:; ~tt::o

Jq. ..

e- ).1;"

= W1.,.. )if

+-r-)I_N-(t)_=_m_]=_V"'l_+_~_rr

IL-E-[N_(_t

_ _ _ _ _ _ _ _ _ _ _ ____.JI

Page 14 of 17

SOLUTION 8. The input, X(t), and output, Y(t), of a linear system are related via the differential

f-H Lt.V) =

equation

2.

I

d2 d d dt 2 Y(t) + 2 dt Y(t) + Y(t) = 2 dt X(t) + 2X(t).

-

r

:l(Jtv) +:Z(i~)-1-l,'~t.~):z.

:z_

.d . . h- l(l.+iW) . . If the mput 1s a zero-mean, w1 e sense statiOnary process w1t corre a.:tiO~n!.__--------..... 2 3 Rx( = (3/4)e -

r)

1rl,

5.z{w):

determine the correlation function of Y(t).

4 +W

..

An answer not supported by appropriate reasoning will not receive any credit.

)

-l'Lt

=-

I'-R-y( - r_)

_=

2.

e

-

4 (+{v2

4 4+t.V"-

- :2.l'Cl

B

z_e_-_''£_1-_e_2_(1_l_____________,l Page 16 of 17