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Unformatted text preview: EEEBSO: Midterm Exam 1 Fall 2001
Examination Date: Oct. 3rd Examination Time: 9140710130 am
Student Name: __§'le0—n Student ID: 1. This exam consists of three problems. 2. One—page crib sheet is allowed. 1. Determine Whether the following statements are TRUE or FALSE (you don’t have to
provide any justiﬁcation for your answer. For each correct answer you will receive 5 points, for each incorrect answer 2 points will be deducted, and no points will be given for no answering. (a) Suppose 31 and B2 are two mutually exclusive events, and A is an arbitrary event.
Then, onmawm=Pman+ﬂAma1
TRuE
(b) Let A and B be two arbitrary events. Then,
PM ﬂ B] 2 PM] + P[B] — 1. TR‘AE (c) Let FX denote the cumulative distribution function of a discrete random vari able X. Then, for any x, FALSE (d) Suppose X and Y are two uncorrelated random variables. Then,
var[X + Y] : var[X] + var[l/]. Time 2. (30’) The probability mass function (PMF) of a random variable X is given as follows: 1/8 3:: —2,
1/4 x: —1,
PX($)= 1/4 x=0,
1/4, a: = 1,
1/8, 33:2. Deﬁne a new random variable Y = X 2. (a) (8’) Find the probability mass function (PMF) of Y. “up, I, 4}
.. yL}. 3=°
PYtj’  3:
)3, l
Kr a==4 (b) (12’) Find the expectation ply and the standard deviation O'y of Y. #3: El:le 459.1 éH—i‘r: 91— . — .‘L
EEY13=¢°+J£W+¢ 16’ 3
1.. 1—1
‘4 4. 7"":: (71': El: Tu ‘ FY (TY : s/‘z. (C) (10’) what is the PM); _ 0'}! g Y 3 ,u + 0y]?
x WW“, 0? = 3/2. “ﬁx” °"Y 5 Y5 ﬁx “7T1 3?[03/2_ éTf: Cd's/2K3 7:” “3/st5 .. _ ._ 3
_ YL3¢°J+PE343 “way? 3. (50’) The joint probability mass function (PMF) of random variables N and K is k 0 otherwise 71. —1(] 107: k=031)I310;n:0117"'
PN‘KU’IJC) = ‘ (a) (15’) Find the marginal PMFS of N and K. o n "to
‘ {0 3 ..—"_F'
Z A}
R30 ‘ _ n ..
: 10 9 {w
n! “(Ln)”: ( \0 (b) (10’) Find the conditional PMF of K given N. F‘Y h'LJJ " Inn) ‘0: ﬂ :0! ‘I  4
thC'I'er
UH“) :
hm! hm)
PI nD o to
: ID :1 ( ‘RJK%)
_____—_______ __ Q‘
‘0" 2“" —
n1
—. ‘0 L \D o 
t? \,K\N(h\n‘) ': {v ( BJKZ‘ It?!" I
o ’.ul.
(C) (15’) Find the expectations ofN and K_
DO (5 iune_l,
ELM]: E n. mm; = “Eu (a m :
=¢ (d) (10’) Are N and K correlated? Show details to justify your answer. for out; chm) 9% Ce. PENCIL») =— FAUI’NU‘) 7? k 0.; N «we Meiersde 5? \< V M «x: uncorrd~*¢d. ‘O ...
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